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Works  of  VICTOR  T.  WILSON 

PUBLISHED   BY 

JOHN   WILEY  &   SONS 


Free-Hand  Perspective. 

For  Use  in  Manual  Training  Schools  and  Colleges. 
By  Victor  T.  Wilson.  8vo,  xii  +  257  pages,  139 
figures.     Cloth,  $2.50. 

Free-Hand  Lettering. 

Being  a  Treatise  cm  Plain  Lettering  from  the 
Practical  Standpoint  for  Use  in  Engineering 
Schools  and  Colleges  8vo.  105  pages,  23  full- 
page  plates.     Cloth,  $1.00. 

Descriptive  Geometry. 

A  Treatise  from  a  Mathematical  Standpoint,  to- 
gether with  a  Collection  of  Exercises  and  Practical 
Applications.  8vo,  viii-l-237  pages,  149  figures. 
Cloth,  $1.50  net. 


DESCRIPTIVE  GEOMETRY 


A    TREATISE    FROM    A    MATHEMATICAL 
STANDPOINT 

TOGETHER   WITH 

A    COLLECTION    OF    EXERCISES    AND 
PRACTICAL  APPLICATIONS 


BY 

VICTOR    T.  WILSON,  M.E, 

PROFESSOR   OF    DRAWING   AND    DESIGN 

MICHIGAN    AGRICULTURAL 

COLLEGE 


FIRST  EDITION 

FIRST    THOUSAND 


NEW  YORK 

JOHN   WILEY   &   SONS 

London:  CHAPMAN  &  HALL,  Limited 
1909 


^ss 


r    L> 


P 


Copyright,  1909 

BY 

VICTOR  T.  WILSON 


COMPOSITION  BY  PRINTED  BY 

RIPLEY   &   GRAY,    PRINTERS  THE    SCIENTIFIC   PRESS 

LANSING,   MICH.  NEW   YORK 


PREFACE. 

Descriptive  geometry  is  essentially  a  mathematical 
subject.  The  application  of  its  principles  to  the  making 
of  working  drawings,  however,  and  the  modifications 
which  are  made  to  suit  the  contingencies  of  practice,  have 
had  a  tendency  to  obscure  this  fact,  and  like  other  theo- 
retical subjects  it  has  suffered  mutilation  in  the  interest  of 
short  cuts  to  immediate  practical  uses.  But  does  not 
technical  education,  after  all,  consist  chiefly  in  an  equip- 
ment of  sound  theory?  It  has  been  the  author's  purpose 
to  refrain  from  any  attempt  to  hold  the  student's  interest 
by  clothing  a  few  principles  with  some  immediate  pratical 
application,  but  instead,  to  present  a  sound  theoretical 
treatment.  How  well  he  has  succeeded  he  leaves  others  to 
judge. 

The  principles  are  herein  formulated  under  theorems, 
as  in  plane  and  solid  geometry;  illustrative  problems  are 
solved  in  accordance  with  these  theorems  and  special 
constructions  discussed.  The  plan  of,'  at  least,  one  well 
know  text  is  followed  of  dividing  all  problems  into  two 
parts ;  the  first  of  which  is  a  statement  of  the  geometrical 
principles  and  the  theoretical  solution  called  an  analysis ; 
the  second  is  a  description  of  the  graphic  solution,  accom- 
panied by  a  drawing.  An  important  feature  is  added, 
however,  of  giving  the  statement  of  the  geometrical  con- 
ditions and  the  solution  in  the  analysis  in  a  general  form, 
instead  of  being  made  to  refer  to  a  certain  kind  of 
problem  exclusively. 


iv  PREFACE 

As  an  illustration  of  the  generalized  treatment 
throughout,  attention  may  be  called  to  the  discussion  of 
the  cone.  A  common  conception  of  a  cone  is  that  of  a 
right  circular  cone,  or  cone  of  revolution.  A  generalized 
definition  is  given,  however,  to  include  all  the  surfaces 
which  may  be  generated  by  a  right  line  moving  so  as  to 
pass  through  a  fixed  point  and  touch  a  curve.  Further,  it 
is  stated  that  every  cone  is  generically  a  right  cone,  and 
may  be  specifically  named  from  the  shape  of  a  cross 
section  so  taken  that  a  perpendicular,  let  fall  from  the 
fixed  point  or  apex  to  the  plane  of  the  section,  will  pierce 
the  latter  in  the  center,  focus,  or  other  characteristic 
point,  as  a  cusp,  point  of  inflection,  etc.,  etc. 

While  the  third  angle  is  undoubtedly  to  be  preferred 
for  working  drawings,  it  is  not  thought  that  descriptive 
geometry,  as  mathematics,  has  any  concern  with  a  partic- 
ular angle.  The  illustrative  problems  used  deal  indiffer- 
ently with  all  so  that  the  emphasis  can  be  laid  upon  sound 
theory. 

Exercises  for  students  are  grouped  in  the  back  of  the 
book  and  suitably  designated  as  belonging  to  a  certain 
part  of  the  text.  An  appendix  deals  with  the  subject  of 
approximate  methods,  which  has  no  proper  place  in  the 
body  of  the  book. 

The  author  wishes  to  acknowledge  his  indebtedness  to 
the  well  known  texts,  notably  those  of  Church  and  of 
McCord,  and  also  in  particular  to  Arthur  G.  Hall,  pro- 
fessor of  mathematics  at  the  University  of  Michigan,  for 
valuable  service  in  examining  the  mathematical  treatment 
of  the  text. 


TABLE  OF  CONTENTS. 


CHAPTEK  I. 

THE  POINT. 
Sections  l  to  12. 

PAGE 

Definition.— The  planes  of  projection.— Lines  and  planes,  not  limited.— 
Parallel  projecting  lines.— Perspective  and  descriptive  geometry.— 
Unfolding  of  the  coordinate  planes.— THEOREM  I.  The  two  project- 
ions of  a  point  fully  determine  the  position  of  the  point  in  space.— 
Axes  of  reference  for  locating  problems.— Exercises. .. 1 


CHAPTEK  II. 

THE  LINE  AND  PLANE. 
Sections  13  to  88. 
The  projections  of  a  plane  defined.—  The  alphabet  of  the  plane.— Notation 
used.— Conventions  recommended.— THEOREM  IL— If  through  anylUne 
in  space,  a  plane  is  passed  perpendicular  to  a  coordinate  plane,  it  will 
Intersect  that  coordinate  plane  in  a  line  which  is  the  projection  of  the 
given  line,  upon  the  coordinate  plane.— Corollaries.— To  locate  a  point 
on  a  line.— THEOREM  HI.- The  projections  of  the  point  of  intersection 
of  two  lines  are  the  points  of  intersection,  respectively,  of  the  project- 
ions of  the  two  lines.— Change  of  coordinate  planes  for  a  point.— The 
trace  of  a  line.— The  trace  of  a  line  on  the  coordinate  planes,  including 
the  end  plane.— To  draw  a  line  in  any  dihedral  angle.— The  alphabet  of 
the  line. — Revolution  of  a  point  about  a  line. — Change  of  one  coordinate 
plane  for  a  line.— To  find  the  true  length  of  a  line _ _ _..ll 


CHAPTER  III. 

PROBLEMS  IN  POINT,  LINE  AND  PLANE. 
Sections  89  to  83. 
THEOREM  IV.  If  a  line  lies  In  a  plane,  the  traces  of  the  line  lie  in  the  cor- 
responding traces  of  the  plane.— THEOREM  V.  A  given  line  is  parallel 
to  a  plane  when  it  is  parallel  to  a  line  in  the  plane,  or  when  it  lies  in  a 
parallel  plane— A  plane  through  three  given  points.— The  traces  of  a 
plane  which  shall  contain  a  given  line.— To  assume  a  point  in  a  plane. 
—THEOREM  VI.  The  traces  of  parallel  planes  with  a  third  plane  are 
parallel.  — Change  of  both  coordinate  planes  with  respect  to  a  line.— 


vi  CONTENTS 

PAGE 

Change  of  coordinate  planes  with  respect  to  a  plane,— A  line  parallel  to 
a  given  plane  through  a  given  point. —  A  plane  through  one  line  and 
parallel  to  another  line.— Within  any  given  plane,  perpendicular  to  a 
coordinate  plane,  to  draw  a  line  making  a  given  angle  with  that  co- 
ordinate plane.— To  pass  a  plane  through  a  given  point  parallel  to  two 
given  lines.— Line  of  intersection  of  two  planes.- Trace  of  a  line  with 
any  plane.— THEOREM  VII.  If  a  line  is  perpendicular  to  a  plane,  the 
projections  of  the  line  are  perpendicular,  respectively,  to  the  traces  of 
the  plane. — Through  a  given  point,  to  pass  a  plane  perpendicular  to  a 
given  line.— Distance  of  a  point  from  a  plane.— Definitions.— THEOREM 
VIII,  If  two  lines  are  perpendicular  to  each  other  in  space,  and  one  of 
them  is  parallel  to  a  coordinate  plane  of  projection,  their  projections 
on  that  plane  are  perpendicular  to  each  other.— The  angle  between  two 
lines  which  intersect.— Projections  of  any  desired  division  of  an  angle 
between  two  lines.— Through  a  point  in  a  plane,  to  draw  two  lines  mak- 
ing a  given  angle  with  each  other.— To  project  a  line  on  any  plane.— The 
angle  a  line  makes  with  a  given  plane.— The  angle  between  two  planes. 
— Given  a  line  in  a  plane,  to  draw  another  plane  intersecting  the  first, 
In  the  given  line,  and  making  a  given  angle  with  the  given  plane.— The 
common  perpendicular  of  two  non-intersecting  lines.  —  Given  one 
trace  of  a  plane,  and  the  angle  the  plane  makes  with  that  coordinate 
plane,  to  find  the  other  trace.— Distance  from  a  point  to  a  line.— Given 
one  trace  of  a  plane,  and  the  angle  it  makes  with  the  corresponding 
plane  of  projection,  to  find  the  corresponding  trace.  —  The  traces  of  a 
plane  making  given  angles  with  both  coordinate  planes,  — Given  the 
angle  a  line  makes  with  both  coordinate  planes,  to  draw  its  project- 
Ions. —To  draw  a  regular  pyramid,  with  base  in  any  given  oblique 
plane.— To  draw  a  circle  through  any  three  given  points.— Through  a 
point  to  draw  a  line  making  a  given  angle  with  a  given  plane.— Through 
a  given  line  to  pass  a  plane  perpendicular  to  a  given  plane.— Loci.— 
Review  questions 84 


CHAPTER  IV. 
GENERATION  AND  CLASSIFICATION  OF  LINES  AND   SURFACES. 
Sections  84  to  98. 
A  line  is  the  path  of  a  moving  point.— Classification  of  lines.— Definitions. 
—Surfaces  as  generated  by  moving  lines.— Classification  of  surface*.— 
Projections  of  curves.— THEOREM  IX.    Two  projections  of  a  curve 
being  given,  the  curve  will,  in  general,  be  completely   determined, 
THEOREM  X.    If  two  lines  are  tangent  in  space,  their  projections  on 
the  same  plane  are  tangent  to  each  other.— A  normal  to  a  curve.— To 
draw  a  tangent  to  an  irregular  curve.- To  find  the  point  of  tangency  of 
an  irregular  curve  and  its  tangent. — Tangent  plane  to  a  surface 83 


CONTENTS  vii 

PAGE 

CHAPTEB  V. 

SINGLE  CURVED  SURFACES. 
Sections  99  to  185. 
THEOREM  XI.  A  plane,  which  contains  two  consecutive  rectilinear 
elements  of  a  single  curved  surface,  will  be  tangent  to  the  surface 
throughout  these  elements,  and  the  converse. — Cylinder  defined.— To 
assume  a  point  on  any  single  curved  surface.— A  tangent  plane  to  a 
cylinder  through  a  point  on  the  surface,  a  point  outside  and  parallel 
to  a  line— A  plane  normal  to  a  cylinder  at  a  point  on  the  surface, 
through  a  point  outside,  parallel  to  a  given  line,  at  a  given  point  on  the 
cylinder  and  parallel  to  a  line.— Elements  of  contour,— The  curves  of 
intersection  of  planes  and  surfaces.— The  development  of  single 
curved  surfaces— The  intersection  of  a  cylinder  and  plane,  and  to 
draw  a  tangent  to  the  curve.— To  develop  a  right  circular  cylinder 
and  curve  of  intersection  of  a  plane— To  develop  an  oblique  cylinder. 
—The  cone.— To  assume  a  point  on  its  surface,  and  to  draw  a  tangent 
plane  at  the  point,  through  a  point  outside,  and  parallel  to  a  line 
outside.— To  draw  a  normal  plane,  at  a  point  on  the  surface,  through 
a  point  outside  and  parallel  to  a  line  outside,  intersection  of  a  cone 
by  any  plane.— To  develop  a  right  cone,  a  cone  In  general— The  con- 
volutes.— The  helix  and  the  helical  convolute.— To  assume  a  point  on 
the  surface  and  to  draw  a  tangent  plane  at  the  point.— Intersection  of 
the  surface  with  any  plane.— Tangent  plane  through  a  point  outside, 
parallel  to  a  line  outside.- To  develop  the  surface _ 98 


CHAPTEE  YI. 

WARPED  SURFACES. 
Sections  136  to  156. 
Warped  surfaces.— Table  of  classification.— The  hyperbolic  parabolold- 
THEOREM  XII.  The  projections  of  all  the  elements  of  one  system  of 
generation  of  a  hyperbolic  paraboloid  upon  the  plane  directer,  inter- 
sect each  other  in  a  point,  known  as  a  'point  of  concourse',  THEOREM 
XIII.  The  section  of  a  hyperbolic  paraboloid,  by  a  plane  parallel  to 
the  two  rectilinear  directrices,  is  a  straight  line.— Corollaries.- To 
assume  a  point  on  the  surface,  to  draw  a  tangent  plane,  at  a  point  on 
the  surface.— To  pass  a  plane  through  a  line  and  tangent  to  it.— Its 
vertex,  axis.— The  conoid.— To  assume  a  point  on  the  surface,  to  pass  a 
plane  tangent  at  the  point.— Right  and  oblique  conoids.— The  right  and 
oblique  helicoid,  to  assume  a  point  on  the  surface,  to  draw  a  tangent 
plane  at  the  point.— Its  intersection  by  a  plane.— THEOREM  XIV.  The 
trace  of  an  oblique  helicoid,  with  any  plane  parallel  to  the  plane 
directer,  is  anarchimedian  spiral _ _ _ 139 


viii  CONTENTS 

PAGK 

CHAPTEE  VII. 

DOUBLE  CURVED  SURFACES  AND  SURFACES  OF  REVOLUTION. 
Sections  157  to  168. 
Double  curved  surfaces.— Tangent  planes.— Laws  of  generation.— Surfaces 
of  revolution.— To  assume  a  point  on  the  surface  and  draw  a  tangent 
plane.— THEOREM  XV.  A  plane  which  is  tangent  to  a  surface  of  re- 
volution at  a  given  point,  is  perpendicular  to  the  meridian  plane 
through  the  point.— Plane  tangent  to  a  sphere.— THEOREM  XVI.  If  two 
surfaces  of  revolution,  having  a  common  axis,  are  tangent  to  each 
other  or  intersect.  It  will  be  in  the  circumference  of  a  circle  whose 
plane  is  perpendicular  to  the  axis  and  center  in  the  axis.— Intersection 
of  surface  of  revolution  with  a  plane.  The  hyperboloid  of  revolution. 
—THEOREM  XVII.  The  hyperboloid  of  revolution,  of  one  nappe,  has 
two  systems  of  generation  and  every  element  of  the  one  system  inter- 
sects all  those  of  the  other  system.—  THEOREM  XVIII,  The  meridian 
curve  of  a  hyperboloid  of  revolution,  of  one  nappe,  is  a  hyperbola.— 
To  assume  a  point  on  the  surface,  and  to  draw  a  tangent  plane  at  the 
point,  its  intersection  with  any  plane,  tangent  plane  through  any  line 
outside - - 165 


CHAPTEK  VIII. 

INTERSECTIONS  OF  SURFACES. 
Sections  169  to  182 
Intersections  of  surfaces.— Intersections  of  bodies  bounded  by  plane  faces. 
—Complete  or  partial  penetration. — Intersection  of  two  pyramids,  two 
cylinders,  two  cones,  cone  and  cylinder,  cylinder  and  convolute,  cone 
and  convolute.— Conditions  of  intersection  of  surfaces  of  revolution.— 
Intersection  of  those  having  a  common  axis,  axes  in  the  same  plane, 
axes  not  in  same  plane.— Intersection  of  a  single  curved  surface  and 
surface  of  revolution.— Intersection  of  cone  or  cylinder  with  a  warped 
surface — __ _ 188 

APPENDIX. 
Discussing  practical  projection.— Third  angle.— Approximate  processes 197 

EXEECISES. 

Series  of  graded  theoretical  problems,  and  also  practical  problems  for 

students  to  solve.    Total  number. _ 199 


DESCRIPTIVE    GEOMETRY 


CHAPTER    I. 


THE    POINT. 

1.  Descriptive  geometry  is  the  science  of  representing 
forms,  plane  and  solid,  by  projecting  them  upon  two  or 
more  planes  at  right  angles  to  each  other  with  the  aid  of 
projecting  lines  perpendicular  respectively  to  these  planes, 
and  it  also  consists  of  the  solution  of  problems  relating  to 
the  properties  and  magnitudes  of  the  forms. 

2.  The    planes  upon    which   objects   are    projected    are 

known  as  the  coordinate  planes  of  projection.  They  con- 
sist, in  general,  of  vertical  plane's,  and  a  horizontal  plane. 
The  vertical  planes  may  be  at  right  angles  to  each  other 
and  are  then  distinguished  as  the  fi'ont  vertical  and  the 
end  vertical  planes  respectively. 

Working  drawings  for  guidance  in  construction  are 
made  according  to  the  general  principles  of  descriptive 
geometry.  The  projection  upon  the  horizontal  plane  cor- 
responds to  the  plan  view  or  simply  plan;  the  projection 
upon  the  front  vertical  plane  corresponds  to  the  front 
view,  front  elevation,  or  simply  elevation;  the  projection 
upon  the  end  vertical  plane  corresponds  to  the  end  view  or 
end  elevation,  as  these  terms  are  variously  used. 


2  DESCRIPTIVE    GEOMETRY 

3.  Lines  and  planes  in  descriptive  geometry,  are  not 
assumed  to  be  limited.  A  line,  which  may  be  designated 
by  letters  at  two  of  its  points,  is  nevertheless  possessed  of 
the  property  of  direction  which  does  not  stop  short  of 
infinity.    A  plane,  likewise,  extends  to  infinity. 

Hence  parallel  lines  are  said  to  have  two  points  in 
common  at  infinity,  and  parallel  planes  to  meet  each  other 
in  a  common  line  at  infinity. 

4.  The  parallel  projecting  lines  spoken  of  in  section  1 
have  a  common  point  at  infinity,  hence  it  is  said  that  the 
''center  of  projection''  is  at  infinity.  This  point  is  also  called 
the  ''point  of  sight  ^*  because,  were  it  possible  for  the  eye 
to  be  in  its  position,  the  forms  projected  from  it  as  a 
center  would  have  their  projections  identical  with  them- 
selves, point  for  point,  line  for  line. 

5.  Perspective  is  a  branch  of  descriptive  geometry  in 
which  the  'center  of  projection'  is  at  a  finite  distance  from 
the  plane  of  projection.  It  gives  the  kind  of  pictures,  for 
example,  that  would  be  seen  if  what  was  beyond  a  window 
were  traced  upon  it  by  a  person  standing  on  the  opposite 
side.  The  eye  is  the  'center  of  projection,'  the  window 
the  'plane  of  projection.' 

6.  Descriptive  geometry  and  perspective  together  are 
parts  of  that  broader  subject  projective  geometry,  in 
which  the  properties  of  figures  are  studied  by  projecting 
them  upon  any  plane  or  planes  from  any  center  of  projec- 
tion and  where  there  is  found  to  be  a  correlation  of  the 


*The  'center  of  projection'  and  'point  of  sight' are  both  equally  good  designing 
terms  used  by  different  authorities. 


THE    POINT 


FIGUBB    1 


properties  of  the  different  projections  of  a  figure  upon  the 
different  planes  and  from  the  different  centers  of  projec- 
tion. 


7.  The  two  fundamental  planes  of  projection  together 
with  two  end  planes  are  shown  in  pictorial  form  or  perspec- 
tive in  Figure  1.  The  front  vertical  and  the  horizontal 
planes  [V  and  H  planes]  form  together  four  dihedral 
angles.  That  which  lies  in  front  of  V  and  above  H  is  the 
1st;  that  which  lies  back  of  V  and  above  H  is  the  2nd,  and 
so  on,  continuing  in  the  same  direction. 

The  line  of  intersection  of  these  two  planes  is  known 
generally  as  the  ground  line  or  abbreviated  the  G.L. 

Objects  are  projected  upon  the  front  vertical  plane  by 
means  of  projecting  lines  such  as  shown  at  v  which  means 
that  the  center  of  projection  for  the  object  is  at  infinity  in 
a  direction  in  front  of  the  vertical  plane,  i.  e.,  opposite  to 


4  DESCRIPTIVE    GEOMETRY 

that  of  the  pointed  end  of  the  arrow.  This  center  may  be 
thought  of  as  either  above  or  below  the  H  plane  as  the 
object  is  above  or  below  it.  It  is  in  fact,  at  infinity  in  the 
horizontal  plane,  and  in  a  direction  perpendicular  to  the 
G.  L. 

Objects  are  projected  upon  the  horizontal  plane  by 
means  of  vertical  projecting  lines  such  as  at  Ji,  whence  the 
center  of  projection  is  considered  as  infinitely  distant 
above  H  in  the  V  plane. 

Objects  are  projected  upon  the  end  planes  by  means 
of  horizontal  projecting  lines  in  the  directions  of  the 
double  headed  arrow  e,  for  the  right  hand  plane  [with 
forms  in  the  first  angle] ,  from  a  center  of  projection  which 
is  at  the  left  and  for  the  left  hand  plane  from  a  center  of 
projection  which  is  at  the  right,  infinitely  distant  in  the  H 
plane. 

8.  The  V,  H  and  end  planes  arc  unfolded  for  the  graph- 
ical representation  of  forms  until  they  form  one  and  the 
same  plane,  see  Figure  2.  Either  V  is  revolved*  about 
the  G.L.  as  an  axis  until  it  coincides  with  H,  or  H  is 
revolved  about  the  G.L.  until  it  coincides  with  V;  the 
result  in  either  case  is  the  same.  The  curved  arrows  in 
Figure  1,  connected  with  the  H  plane,  show  the  direction 
of  the  revolution. 

The  end  plane,  which  is  at  the  right  of  any  object, t  is 
revolved  about  the  ground  line  with  the  vertical  plane  or 


*An  object  Is  said  to  'rotate'  upon  Its  own  axis  and  to  'revolve'  about  any 
other  axis  not  its  own. 

t  In  the  theoretical  treatment  of  the  subject. 


THE    POINT 


/ 

E  plane  above 
H  and  in 
front  of  V 

V  above  H 
H  bacl£  of  V 
E  plane  above                              E  plane  above 
H  and  back                                   H  and  back 
of  V                                                of  V 

"^ { 

E   plane  above      \ 
H   and  in           I 
front  of  V 

L. 

r 

\     E  plane  below 
)        H  and  in 
1        &X)ntofV 

E  plane  below                             E  plane  below 
H  and  bock                                 H  and  back 
of  Y              H  In  Front  of  V     of  V 
V  below  H 

Le 

E  plane  below       / 
H  and  in            ( 
front  of  V           \ 

FlQUBB    2. 

GbLb  until  it  lies  in  V,  that  part  which  is  in  front  of  V 
going  toward  the  right,  that  part  which  is  back  of  V  going 
toward  the  left.  Similarly,  the  end  plane,  which  is  at  the 
left  of  any  object,  is  revolved  into  coincidence  with  V,  the 
portion  which  is  in  front  of  V  going  towards  the  left. 


9.  Let  A  be  any  point  in  space  in  front  of  V  and  above 
H,  i.e.,  in  the  first  dihedral  angle.  Conceive  a  plane  to 
be  passed  through  it  perpendicular  to  both  H  and  V.  It 
would  correspond  to  an  end  vertical  plane.  This  plane 
would  cut  V  in  a  vertical  line  and  H  in  a  horizontal  line, 
both  perpendicular  to  the  G.L.  at  the  same  point.  When 
V  and  H  are  unfolded,  these  lines  would  become  one  and 
the  same  straight  line  perpendicular  to  the  G.L. 

The  point  A  is  projected  on  V  by  a  projecting  line 
perpendicular  to  V  and  lying  in  this  end  plane.  It  is  a 
line  parallel  to  H.  A  is  projected  on  H  by  a  projecting 
line  perpendicular  to  H,  also  lying  in  this  end  plane.  It  is 
a  line  parallel  to  V.    The  piercing  points  of  these  projecting 


6  DESCRIPTIVE     GEOMETRY 

lines  with  the  coordinate  planes  constitute  the  respective 
projections  of  A  on  those  planes.  From  whence:  The 
two  projections  of  a  point  are  on  a  common  perpendicular  to 
the  G,L. 

When  the  Y  and  H  planes  are 
unfolded,  see  Figure  3,  a  is  the 
projection  on  H  of  the  point  A 
and  oa  is  the  projection  on  H  of 
the  projecting  line  of  A  on  V.  oa 
is  equal  to  the  distance  of  the 
point  A  from  V.  Also  a'  is  the 
projection  on  V  of  the  point  A 
and  oa'  is  the  projection  on  V  of 
the  projecting  line  of  A  on  H,  and 
oa'  is  the  distance  of  the  point  A 
from  H.    This  is  true  for  a  point 

in  either  dihedral  angle  or  when  lying  in  either  or  both 

coordinate  planes. 

10,  Descriptive  geometry  does  not  deal  with  forms 
themselves  but  with  the  projections  of  the  forms.*  Fig.  4 
illustrates  the  projections  of  a  point  when  variously  placed 
with  respect  to  the  coordinate  planes.  A  is  in  the  1st 
angle;  B  is  in  the  2nd;  C  is  in  the  3rd;  D  is  in  the  4th; 
E  is  in  V  above  H;  F  is  in  V  below  H;  Gr  is  in  H  in  front 
of  V;  I  is  in  H  back  of  V;  J  is  in  the  G.  L.  whence  both 
projections  coincide  with  the  point  itself.    When  the  pro- 


FI QUBE    8 


♦Hence  the  words 'the  projection  of  are  unnecessary  and  when  hereafter  a 
form  is  spoken  of,  it  means  the  projection  of  the  form. 


THE  POINT 


«> 


I 

I 
I 


0' 


33     L. 


dl 


s% 


c'i 


di 


FlGTJBB    4. 


jections  and  the   forms   happen   to  coincide,  it   avoids 
confusion  to  consider  the  projections  only. 

The  positions  shown  in  Figure  4  are  known  as  the 
alphabet  of  the  point,  being  the  total  of  the  various 
positions  a  point  can  have  relative  to  the  coordinate  planes. 

11  THEOREM   I. 

The  two  projections  of  a  point  fully  determine  the  posi- 
tion of  the  point  in  space. 

Proof; — If  at  the  V  projection  of  the  point,  a  perpen- 
dicular is  erected  to  the  V  plane,  by  hypothesis  it  will 
pass  through  the  point  in  space.  If  at  the  H  projection 
of  the  point,  a  perpendicular  is  erected  to  the  H  plane, 
it  will  also  pass  through  the  point  in  space.  Hence  the 
point  itself  must  lie  at  the  intersection  of  these  two 
perpendiculars.  But  the  perpendiculars  are  the  two 
projecting  lines  of  the  point  on  the  coordinate  planes. 
Hence  the  point  is  fully  determined. 


8  DESCRIPTIVE     GEOMETRY 

If  the  point  is  in  the  H  plane,  the  length  of  the  per- 
pendicular to  the  H  plane  is  zero,  and  the  point  is  its  own 
projection.  Likewise,  if  the  point  is  in  the  V  plane,  it  is 
its  own  V  projection.  If  the  point  is  in  the  G.L.,  both 
perpendiculars  are  zero  and  it  is  its  own.V  and  H  pro- 
jection. 

12.  Axes  of  reference  may,  for  convenience  in  solving 
problems,  be  taken  to  coincide  with  the  lines  oa  and  oa' 
of  Figure  3,  the  line  oa',  as  a  portion  of  a  i/  axis  and  oa 
as  a  portion  of  an  x  axis.  Hence,  the  position  of  the 
point  A  can  be  specified  numerically  by  coordinates.  For 
example,  referring  to  Figure  3,  A  =  1,  |,  meaning  that  it 
is  1  inch  above  H  and  f  inch  in  front  of  V,  or  its  V  projec- 
tion is  1  inch  above  the  G.L.  and  its  H  projection  is  f  of 
an  inch  in  front  of  the  G.L. 

The  G.L.  may  also  be  taken  as  a  ^  axis,  being  perpen- 
dicular to  the  plane  of  the  x  and  y  axes  at  their  intersec- 
tion or  origin,  so  that  the  exact  location  of  points  on  a 
drawing  can  be  specified  by  three  coordinates,  z,  y  and  x^ 
given  in  the  order  named ;  the  z  can  be  a  distance  from 
any  convenient  reference  point,  preferably  one  to  the  left 
as  the  border  line  of  a  sheet  of  drawings,  the  2/  as  +  or  — 
according  as  the  point  is  above  or  below  H  and  x  as  -\-  or 
—  according  as  the  point  is  in  front  of  or  behind  V. 

For  example,  in  Figure  4,  C  can  be  specified  as  C  = 
1,  —  1t6,  —  1,  where  1  means  the  z  measured  from  the 
dotted  line  connecting  a'  and  a;  A  can  be  specified  as  A 
=  0,  i,  f.  The  other  points  specified  similarly,  are:  B 
=  i,  1,  -  h  D  =  If,  -  i,  1;  E  ==  II,  i  0;  F  =  2,  -  i,  0; 
G  =  2i,  0,  f ;  I  =  2f ,  0,  -  f ;  J  =  3i,  0,  0. 


THE  POINT  9 

When  the  z  coordinate  for  any  point  is  given  as  zero, 
let  it  be  understood  that  the  xy  plane  for  the  point  may  be 
taken  anywhere,  and  the  other  points  following  are  then 
measured  from  this  plane  as  origin. 

If  either  the  y  or  the  x  coordinate  of  any  point  is  miss- 
ing and  in  its  place  is  a  question  mark  or  a  dash,  it 
indicates  that  the  coordinate  omitted  is  to  be  found 
through  other  data  in  the  problem. 

Exercises. 

1 
Locate  the  following  points,  designating  the  pro- 
jections by  small  letters,  as  a'  for  the  V  projection  and  a 
for  the  H  projection,  and  state  how  each  is  related  to  the 
coordinate  plane  of  projection:  A  =  0,  1,  —\\  B  =  J,  — 
i,  -1;  C  =  1,  l,i;  D  -  U,  -1,  1;  E  =  2,  1,  0;  F  =  2i,  0, 
-1;  0  =  3,0,1;  1  =  31,0,0. 

2 

Locate  the  following  points:  A  =  0,  f,  ^;  B  =  i, 
-  i  -  i;  C  =  4,  f,  -  i;  D  =  i,  -  i,  f ;  F  =  f,  0,  f;  G  = 
f,  I,  0;  I  =  li,  0,  0.  Project  the  same  upon  an  end  ver- 
tical plane  at  the  right  whose  GbLe  cuts  the  Gr.L.  at  the 
point  0  =  2i,  0,  0. 

Construction.  Figure  5  shows  the  result.  Since  the  end 
plane  is  perpendicular  to  V  and  H,  the  point  A  is  projected 
upon  it  by  means  of  a  projecting  line  parallel  to  V  and  H. 
This  projecting  line  will  pierce  the  end  plane  in  a  point 
which  is  as  far  from  V  and  H  as  the  point  A  is  distant  from 
those  planes,  i.  e.,  as  far  from  the  ground  line  in  each  case 
between  the  Y,  the  H,  and  the  end  planes.  When  the  end 
plane  is  folded  into  coincidence  with  the  V  plane,  the 
projection  of  A  upon  it  is  a  point  at  a  distance  oa  from  the 


10 


DESCEIPTIVE     GEOMETRY 


Figure  5. 

G-K  Le  and  a  distance  oa'  above  the  G.L.  The  G.L.  is 
coincident  with  the  revolved  position  of  the  ground  line 
between  the  end  plane  and  the  H  plane.  That  which  is 
at  the  left  of  Ge  Le  is  that  portion  of  the  end  plane  which 
lies  beyond  the  V  plane.  So  when  considering  the  pro- 
jections upon  the  end  plane  alone,  Ge.  Le  is  the  V  plane  in 
end  projection  and  the  G.L.  is  the  H  plane  seen  in  end 
projection  so  formed  by  looking  at  the  end  plane  toward 
the  right  in  the  direction  of  the  G.L.  of  the  two  coordinate 
planes. 

Likewise  the  end  projection  of  the  point  B  is  found  at 
a  distance  oh'  below  the  G.L.  and  a  distance  oh  to  the  left 
of  the  Ge  Lb  ,  being  in  the  3rd,  dihedral  angle,  and  so  on 
for  the  other  points. 

3. 

Draw  the  projections  of  the  following  points:  A, 
1' '  below  Hand  I' '  in  front  of  V;  B,  in  the  3rd  dihedral 
angle  W  '  from  both  V  and  H;  C,  U"  behind  V  and  f  ^ 
below  H;  D,  in  H,  If '  behind  V;  E,  in  V,  If '  below  H. 
Project  these  also  upon  an  end  plane  whose  Ge.  Lb.  may 
be  chosen  at  pleasure. 


CHAPTER  II 


THE  LINE  AND  PLANE. 

13.  A  plane  is  projected  upon  the  coordinate  planes  only 
by  the  projection  of  points  or  lines  lying  in  it  or  by  draw- 
ing its  lines  of  intersection  with  the  coordinate  planes, 
designated  generally  as  traces,  and  specifically  the  V  and 
H  traces. 

Since  a  plane  can  intersect  a  line  in  only  one  point, 
the  H  and  V  traces  of  a  plane  will  intersect  upon  the  G.L. 

If  a  plane  is  oblique  to  H  and  V  and  also  to  the  G.L., 
its  H  and  V  traces  are  inclined  to  the  G.L.,  meeting  it  in 
a  finite  point. 

If  a  plane  is  parallel  to  the  Gr.L.  but  oblique  to  H 
and  Y,  its  traces  are  parallel  to  the  G.L.  and  intersect  the 
latter  at  infinity. 

If  a  plane  is  parallel  to  V  its  H  trace  is  parallel  to 
the  G.L.  and  its  V  trace  is  at  infinity.  Similarly,  if  a  plane 
is  parallel  to  H,  its  Y  trace  is  parallel  to  the  G.L.  and  its 
H  trace  is  at  infinity. 

If  a  plane  passes  though  the  G.L.  then  both  of  its 
traces  coincide  with  the  G.L.,  and,  referred  only  to  its 
traces,  the  plane  is  indeterminate. 

Figure  6  shows  the  several  positions.    1  and  la  are 


12  DESCRIPTIVE     GEOMETRY 

.      / 


/ 
^/      \o,-  RV 


SV  4 


<"/  ^^9^ 


7 


2  UV  5        UH     L 


\ 


Figure  6. 


the  traces  of  a  plane  oblique  to  H  and  V  and  to  the  G.L. 
2  is  a  plane  parallel  to  the  G.L.  3  is  a  plane  parallel  to  V. 
4  is  a  plane  parallel  to  H.  5  is  a  plane  passing  through 
the  G.L. 

14.  The  alphabet  of  the  plane  is  an  expression  used  to 
designate  the  possible  positions  of  a  plane  with  respect  to 
the  coordinate  planes.    There  are  thirteen  such  positions. 

(1).  A  plane  oblique  to  the  G.L.  as  in  1  or  la  of 
Figure  6. 

(2) .  A  plane  parallel  to  the  G.L.  and  cutting  through 
the  1st,  2nd  and  3rd  dihedral  angles. 

(3) .  A  plane  parallel  to  the  G.L.  and  cutting  through 
the  1st,  4th  and  3rd  dihedral  angles. 

(4) .  A  plane  parallel  to  the  G.L.  and  cutting  through 
the  2nd,  1st  and  4th  dihedral  angles. 

(5) .  A  plane  parallel  to  the  G.L.  and  cutting  through 
the  2nd,  3rd  and  4th  dihedral  angles. 

(6) .    A  plane  perpendicular  to  H  and  oblique  to  V. 

(7) .    A  plane  perpendicular  to  V  and  oblique  to  H. 


THE    LINE   AND    PLANE  13 

(8) .      A  plane  perpendicular  to  both  H  and  V. 

(9).      A  plane  passing  through  the  G.L. 

(10) .  A  plane  parallel  to  V  and  passing  through  the 
1st  and  4th  dihedral  angles. 

(11).  A  plane  parallel  to  V  and  passing  through  the 
2nd  and  3rd  dihedralangles. 

(12).  A  plane  parallel  to  H  and  passing  through  the 
1st  and  2nd  dihedral  angles. 

(13).  A  plane  parallel  to  H  and  passing  through  the 
3rd  and  4th  dihedral  angles. 

15.  The  following  notation  will   be  followed  throughout 

the  text  and  in  the  exercises  at  the  end  of  the  book. 

A  point  in  space  is  designated  by  a  capital  letter,  as 
A,  its  projections  by  a  small  letter;  if  the  V  projection  a' 
and  the  H  projection  simply  a;  if  the  end  projection  ae. 

A  line  is  specified  in  general  by  two  points  on  the  line. 

A  plane  is  designated  by  lettering  its  traces  with  a 
capital  letter  followed  by  Y,  H  or  E  according  as  it  is  the 
vertical  trace,  horizontal  trace,  or  a  trace  with  an  end 
plane;  the  last  letters  of  the  alphabet  will,  in  general,  be 
used  for  this  purpose.  The  letter  does  not  specify  any 
point  on  the  trace. 

The  line  of  intersection  of  the  V  and  H  planes  is  called 
the  O.L.  The  line  of  intersection  of  V  or  H  and  an  end 
plane  is  called  the  Ge.Le. 

16.  The  following  line   conventions   are  recommended: 

The  two  projections  of  a  point  are  connected  by  a  dotted 
line,  the  dots  to  be  about  iV  inch  long,  with  A  inch 
spaces  between  the  dots. 

The  G.L.  and  given  lines,  light,  solid. 


14  DESCRIPTIVE    GEOMETRY 

The  Required  lines,  heavy,  solid. 

The  projecting  lines,  construction  and  hidden  edges 
of  solids,  dotted  lines. 

Auxiliary  lines,  dash  with  dashes  about  i  inch  long 
and  iV  inch  spaces  between  dashes. 

Traces  of  given  planes,  dash  and  dot;  \  inch  dashes 
with  T6  inch  dots  and  sV  inch  spaces. 

Traces  of  required  planes,  the  same,  heavy. 

Axes  of  revolution  and  axes  of  solids,  dash  and  two 
dots,- 

It  will  be  found  helpful  in  the  earlier  part  of  the  work 
to  show  all  parts  of  forms  in  the  1st  angle  as  solid  lines 
and  those  in  the  other  angles  dotted. 

17.  THEOREM  II. 

If  through  any  line  in  space  a  plane  is  passed  perpendic- 
ular to  a  coordinate  plane,  it  will  intersect  that  coordinate 
plane  in  a  line  which  is  the  projection  of  the  given  line 
upon  the  coordinate  plane. 

Proof: — For  this  projecting  plane,  by  definition  of  pro- 
jection, contains  the  projecting  perpendiculars  of  all 
points  of  the  line  upon  the  coordinate  plane,  the  locus  of 
their  intersections  with  the  coordinate  plane  constitutes 
the  trace  of  the  plane  upon  that  coordinate  plane. 

18.  Corollary  1.  A  line  is,  in  general,  fully  determined 
when  its  projections  are  given,  because  if,  through  one  of 
the  projections  of  the  line,  a  plane  is  passed  perpendicular 
to  that  coordinate  plane,  it  will  contain  the  line,  and  if 
through  the  corresponding*  projection  a  plane  is  passed 

♦Note  the  use  of  the  word  'corresponding'  occurring  throughout  l"he  book; 
Jt  facilitates  the  generalized  treatment,  makes  processes  applicable  to  either 
coordinate  plane. 


THE    LINE    AND    PLANE  15 

perpendicular  to  the  corresponding  coordinate  plane,  it 
will  also  contain  the  line.  Hence  the  line  itself  must  be 
identical  with  that  line  which  is  common  to  both  planes, 
namely  their  intersection. 

19.  Corollary  2.  If  the  projections  of  two  points  upon 
one  of  the  coordinate  planes  are  connected  by  a  line,  it  is 
that  projection  of  the  line  joining  the  two  points  in  space. 
Similarly,  if  the  corresponding  projections  of  the  points 
are  connected  by  a  line,  it  is  the  corresponding  projection 
of  the  line  connecting  the  two  points  in  space. 

A  line  connecting  the  projections  of  two  points  on  a 
coordinate  plane  is  indeterminate  if  it  lies  in  an  end  plane 
for  the  two  projecting  planes  of  the  line  coincide. 

20.  To  designate  a  point  on  any  line,  assume  either  pro- 
jection of  the  point  upon  the  same  projection  of  the  line, 
the  other  or  corresponding  projection  is  found  upon  the 
corresponding  projection  of  the  line,  at  the  intersection 
with  it,  of  a  perpendicular  to  the  G.L.  through  the 
assumed  projection  of  the  point. 

21.  The  projecting  planes  of  a  line  are  useful  in  the 
solution  of  problems.  Since  the  coordinate  planes  are 
perpendicular  to  each  other  and  the  projecting  plane  of  a 
line  is  perpendicular  to  one  of  them,  it  follows  that  its 
corresponding  trace  is  a  line  perpendicular  to  the  G.L. 

Figure  7  shows  the  projections  of  a  line  AB.  SH  and 
SVare  the  traces  of  its  H  projecting  plane;  TH  and  TV 
are  the  traces  of  its  Y  projecting  plane.  That  part  of  SV 
which  is  above  the  G.L.  is  the  intersection  of  S  with  Y 


16 


DESCRIPTIVE    GEOMETRY 


Figure  7. 

above  H  and  that  part  which  is  below  the  G.L.  is  the 
intersection  of  S  with  V  below  H.  Similarly,  that  part  of 
the  H  trace  which  is  in  front  of  the  G.L.  is  the  intersection 
of  S  with  H  in  front  of  V*  and  that  part  of  the  H  trace 
which  is  back  of  the  G.L.,  is  the  intersection  of  S  with  H 
back  of  y. 

The  traces  of  a  plane  are  not  limited  by  the  G.L, 

22.  THEOREM  III. 

The  projections  of  the  point  of  intersection  of  two  lines 
are  the  points  of  intersection  respectively  of  the  projections 
of  the  two  lines. 

Proof: — The  point  of  intersection  of  two  lines,  being 
common  to  both,  can  have  but  one  projection  on  V  and 
one  projection  on  H.  And  since  the  point  lies  on  both 
lines  it  must  lie  on  the  projection  of  both  lines;  the  only 
point  which  satisfies  these  conditions  is  the  point  of 
intersection  of  the  projection  of  the  lines. 


*It  Is  well  to  speak  of  the  H  projection  of  a  point  as  in  front  of  the  G.L.,  not 
'below,'  also  back  of  and  not '  above.' 


THE    LINE    AND    PLANE 


17 


23.  The  form  of  proof  to  theorems,  in  descriptive 
geometry,  differs  in  some  respects  from  that  nsed  in  math- 
ematical subjects  heretofore  studied  by  the  student,  and  a 
failure  to  readily  grasp  the  significance  or  meaning  of  the 
various  steps  is  a  common  source  of  difficulty.  As  in 
other  mathematics,  each  step  in  descriptive  geometry  is 
capable  of  demonstration  and  proof,  and  should  be  so 
treated  in  all  study.  The  proofs  of  steps  should  be  followed 
mentally  until  the  processes  involved  are  performed  uncon- 
sciously. A  faithful  adherance  to  this  direction  will  re- 
jnove  much  of  the  difficulties  in  the  study  of  the  subject. 


\  i'  ?  1  >r 

\\  1    1/  Y 

G.              '\     ^j^^                      ! 

Figure  8. 

24.    The  change  of  a  coordinate   plane   is  many    times 
useful.    Consider  a  change  of  the  V  coordinate  plane. 

Let  A,  B  and  C  be  three  points  shown  in  projection 
in  Fig.  8.  Project  the  same  upon  a  new  V  plane  or  Vi,  cut- 
ting H  in  the  line  Gr  i  .L  ^ .  The  Vi  plane  is  revolved  about 
the  Gti  .Li .  into  coincidence  with  H,  that  which  is  above  H 
going  downward  toward  the  left.    The  two  projections  of  a 


18  DESCRIPTIVE     GEOMETRY 

point  are  on  a  common  perpendicular  to  the  G.L.,  hence 
A  is  projected  on  a  perpendicular  to  the  Gi  .Li .  through  a. 
The  distance  of  A  above  H  remains  the  same,  hence  a/  is 
a  distance  from  Gi.Li.  that  a'  is  above  the  Gr.L.  The 
point  B  is  in  the  2nd  angle,  but  Gi  .Li .  cut  H  back  of  h, 
therefore,  when  6/  is  located  on  a  perpendicular  to  the 
Gi  .Lj .  at  a  distance  above  it  that  h'  is  above  the  G.L.,  it 
brings  the  point  B  in  the  1st  dihedral  angle  with  respect  to 
V 1 .  C  lies  in  V  and  its  H  projection  is  at  the  intersection 
of  the  Gi  .L 1 .  with  the  G.L.,  hence,  it  also  lies  in  the  new 
V,  plane  at  c'  a  distance  from  c  that  c'  if  from  c. 

The  new  H  projections  do  not  change;  the  distances 
of  points  from  H  are  constant,  hence  the  distances  of  their 
new  projections  in  front  of  or  back  of  the  new  Gi.  Li.  are 
constant. 

25.  A  new  coordinate  plane,  or  Hi  plane,  may  be  used 
with  respect  to  either  V  or  Y^.  This  is  not  a  horizontal 
plane  but  is  called  an  Hi  plane  for  convenience  and  to 
associate  it  with  the  V  or  Vi  planes.  Let  the  ground  line 
of  the  Hi  and  Y  or  Vi  planes  be  called  G2:.  Lg .  The  Hj 
projection  of  a  point  is  found  upon  a  perpendicular  to  the 
G2.  L2.  through  the  Y  projection  and  at  a  distance  from 
G2.  L2.  that  the  H  projection  is  from  the  G.  L.  Or  the  Hi 
projection  is  found  upon  a  perpendicular  to  the  G2.  L2. 
through  the  Yi  projection  and  at  a  distance  from  G2.  L2. 
that  the  H  projection  of  the  point  is  from  the  Gi.  Li. 

In  this  change  the  Y  or  the  Yi  projection  does  not 
change,  nor  the  distance  of  the  point  from  the  Y  or  Yi 
plane.  The  distance  of  a  point  from  the  Hi  plane  may  be 
different  from  it  distance  from  H. 


THE    LINE    AND    PLANE 


19 


FIGUBB    9. 

Let  A  and  B  be  two  points  in  projection  in  Figure  9. 
Project  them  also  upon  a  Vi  plane  whose  Gi.Li.  cuts  the 
G.L.  as  shown.  A  new  Hi  plane  also  cuts  Vi  in  a  line 
G2.  Lg.  at  right  angles  to  the  Gi.  Li.  a^'  is  found  at  a 
distance  above  the  Gi.Li  that  a'  is  above  the  G.  L.  and 
«!  is  found  on  a  perpendicular  to  the  G2.L2.  through  ^i' 
and  at  a  distance  from  it  that  a  is  from  the  Gi.Li.  Simi- 
larly &^  is  a  distance  in  front  of  the  Gi  L^  ,  i.  e.,  to  the  left, 
that  B  is  from  the  H  plane  shown  Sith\  and  6.  is  a  distance 
in  front  of  G2  L2  that  h  is  from  the  Gi  Li  . 

26.    If  a  line  is  parallel  to  either  coordinate  plane,  the 

trace  of  the  projecting  plane  of  the  line  on  the  other  or 
corresponding  plane  of  projection  is  a  line  parallel  to  the 
G.L. 


If  a  line  is  parallel  to  both  coordinate  planes  of  projec- 
tion then  the  traces  of  both  projecting  planes  of  the  line 
are  parallel  to  the  G.L.    Hence,  when  a  line  is  parallel  to 


20  DESCKIPTIVE    GEOMETRY 

y  its  H  projection  is  parallel  to  the  G.L.  and  when  it  is 
parallel  to  H  its  V  projection  is  parallel  to  the  G.L.  If  it 
is  parallel  to  both  H  and  V  it  is  parallel  to  the  G.L.  and 
both  projections  are  parallel  to  G.L. 

If  a  line  is  parallel  to  a  plane  all  points  of  it  are 
equally  distant  from  that  plane,  hence  the  projection  of  a 
line  on  a  coordinate  plane  to  which  it  is  parallel  will  show 
the  true  length  of  the  line,  and,  by  its  angle  with  the  G.L., 
the  angle  the  line  makes  with  the  corresponding  plane  of 
projection. 

If  a  line  is  perpendicular  to  a  coordinate  plane  its 
projection  on  that  plane  is  a  point  and  its  projection  on 
the  corresponding  coordinate  plane  is  a  line  perpendicular 
to  the  G.L.  and  equal  in  length  to  the  line  itself. 

If  a  line  is  parallel  to  an  end  plane,  it  is  necessary  to 
project  the  line,  by  two  of  its  points  upon  the  end  plane, 
to  determine  its  length,  direction,  and  its  angles  with 
either  coordinate  plane. 

27.  The  trace  of  a  line  is  the  point  of  intersection  of 
the  line  with  any  plane.  Commonly,  in  descriptive  geom- 
etry, it  means  the  point  of  intersection  of  a  line'  with  one 
of  the  coordinate  planes  of  projection. 

Eef erring  to  Fig.  10,  every  point  in  the  vertical  pro- 
jection of  A  B,  being  the  vertical  projection  of  some  point 
in  the  line,  shows  that  the  point  D  in  which  this  projection 
cuts  th^  G.  L.  is  the  vertical  projection  of  the  point  of  the 
line  which  is  in  H*,  that  is,  it  is  the  vertical  projection  of 


*  The  G.  L.  contains  the  vertical  projections  of  all  points  lying  in  the  H 
plane,  in  fact  it  is  the  vertical  projection  of  the  H  plane.  It  also  contains  the  H 
projections  of  all  points  in  the  V  plane,  and  it  is  likewise  the  H  projection  of  the 
V  plane. 


THE    LINE    AND    PLANE 


21 


Figure 


the  H  trace  of  the  line  AB.  The  H  projection  of  the  H 
trace  is  on  the  H  projection  of  the  line.  Similarly  the 
point  of  intersection  of  the  H  projection  of  AB  with  the 
G.  L.,  e,  is  the  H  projection  of  the  V  trace;  the  V  project- 
tion  of  the  V  trace  is  on  the  V  projection  of  the  line. 
Hence  may  be  derived  the  following: 


28.  To  find  the  trace  of  a  line  with  either  coordinate 
plane,  prolong  the  corresponding  projection  of  the  line 
until  it  cuts  the  G.  L.  It  is  the  corresponding  projection 
of  the  trace  desired.  The  other  projection  of  the  trace  will 
be  found  upon  the  other  projection  of  the  line. 

The  preceding  is  a  general  statement.  It  may  be  di- 
vided more  specifically  into  a  rule  for  finding  the  H  trace 
of  a  line  and  one  for  finding  the  V  trace  of  a  line.  The 
first  of  these  can  be  stated  as  follows : 

Prolong  the  Y  projection  of  the  line  until  it  intersects 
the  G.  L.,  which  point  is  the  V  projection  of  the  H  trace; 
the  H  projection  of  the  H  trace  is  on  the  H  projection  of 
the  line  at  the  intersection  of  a  perpendicular  to  the  G.  L. 
through  the  V  projection. 

Observe  that  neither  of  the  two  coordinate  points  alone 
is  the  trace  sought,  but  the  two  together  constitute  the 
projections  of  the  trace. 


22 


DESCRIPTIVE    GEOMETRY 


Figure    11. 

29.  Let  AB  and  CD,  in  Fig.  1 1,  be  two  intersecting  lines 

the  first  of  which  is  parallel  to  H  and  the  second  parallel  to 
V.  The  point  E  is  the  V  trace  of  the  line  AB.  The  H 
trace  is  at  infinity.  The  V  projection  of  the  H  trace  will 
be  on  the  G.  L.  at  infinity  and  the  H  projection  of  the  H 
trace  will  be  on  the  H  projection  of  the  line  at  infinity. 
Similarly  S  is  the  H  trace  of  the  line  CD.  The  H  projec- 
tion of  the  y  trace  is  on  the  G.L.  at  infinity,  the  V 
projection  of  the  V  trace  is  on.  the  V  projection  of  the  line 
at  infinity. 

Hence  to  make  a  general  statement :  When  a  line  is  par- 
allel to  either  coordinate  plane  the  projection  of  its  trace 
on  that  plane  is  on  that  projection  of  the  line  at  infinity; 
the  corresponding  projection  of  the  trace  is  on  the  G.  L. 
at  infinity. 

30.  If  a  line  is  parallel  to  both  coordinate  planes  its  H 
and  V  traces  are  identical  both  upon  the  G.  L.  and  the 
line  at  infinity. 

If  a  line  goes  through  the  G.  L.  at  a  finite  point,  that 
point  constitutes  both  projections  of  both  traces. 


THE    LINE    AND    PLANE 


23 


FlQUBB     12. 


If  a  line  lies  in  an  end  plane  its  traces  are  in  the 
ground  line  or  Ge  .  Lb  .  of  the  end  plane  with  the  Y  or  H 
plane.  The  rule  is  modified  to  suit  the  condition.  For  ex- 
ample see  Fig.  12.  AB  is  a  line  in  an  end  plane.  Eevolve 
this  plane  about  its  Ge  .Le  .  into  the  V  plane  and  get  the  end 
projection  of  AB  at  ae  he .  To  obtain  the  H  trace  of  the 
line  prolong  the  end'  projection  of  the  line  until  it  inter- 
sects the  G.  L.  which  is  the  revolved  line  of  intersection  of 
the  end  plane  with  the  H  plane.  Let  Se  be  the  end  projec- 
tion of  the  H  trace.  The  H  projection  of  the  H  trace  will 
be  on  the  H  projection  of  the  line  at  a  distance  above  the 
G.  L.  that  Se  is  to  the  left  or  back  of  the  Ge  .  Lb  .  The  V 
projection  of  S  will  be  on  the  G.  L.  at  its  intersection  with 
the  Ge  .  Lb  .  Similarly  to  obtain  the  V  trace  prolong  the 
end  projection  of  the  line  until  it  intersects  the  Ge  Lb 
which  is  in  y.  The  H  projection  of  the  V  trace  is  in  the 
G.  L.  at  the  intersection  with  Ge  .  Le  . 


24  DESCRIPTIVE     GEOMETBY 

31.  To  find  the  traces  of  a  line  by  means  of  an  end  plane, 

project  the  line  upon  the  end  vertical  plane,  then  prolong 
this  projection,  if  necessary,  until  it  intersects  the  V 
plane  in  the  Ge.Lk.,  and  the  H  plane  in  the  G.L.  The 
former  is  the  V  projection  of  the  V  trace,  the  latter  is  the 
revolved  H  projection  of  the  H  trace.  The  H  projection 
of  the  V  trace  is  in  the  G.  L.  and  the  H  projection  of  the  H 
trace  is  on  the  H  projection  of  the  line,  a  distance  in  front 
of  or  back  of  the  G.  L.  that  the  point  is  in  front  of  or  back 
ofV. 

32.  The  proper  study  of  descriptive  geometry  requires 
that  every  problem  be  divided  into  two  parts,  first  analysis, 
second,  solution  or  graphic  demonstration.  The  first  is 
the  consideration  of  the  geometrical  principles  involved, 
together  with  a  description  of  the  method  of  solution  if  it 
is  a  problem  or  of  laying  out  the  data  if  it  is  simply  the 
graphic  interpretation  of  the  geometric  conditions;  its 
treatment  should  be  general.  The  second  concerns  the 
graphical  part  and  consists  in  solving  the  problem  graph- 
ically according  to  the  principles  decided  upon  in  the  ana- 
lysis or  graphically  interpreting  the  data  if  it  be  only  the 
interpretation  of  descriptive  geometry  conditions. 

The  student  is  cautioned  never  to  hasten  to  the  second 
without  doing  full  justice  to  the  first.  The  analysis  may 
involve  the  statement  of  descriptive  geometry  conditions, 
the  geometric  principles  involved  and  a  division  of  the 
main  problem  into  parts  each  of  which  has  its  separate 
solution.  However  simple  or  complex  the  problem  is,  it 
should  have  its  analysis  preceding  the  graphical  interpre- 
tation. 

Verify  geometrically  all  steps  or  processes  by  going 


THE    LINE    AND    PLANE  25 

over  the  most  fundamental  and  elementary  principles  in- 
volved in  each  problem  until  the  interpretation  of  the 
fundamentals  no  longer  gives  any  trouble,  but  is  dealt  with 
as  unconsciously  as  letters  are  formed  to  spell  words  in 
writing.  The  entire  subject  depends  upon  a  few  compara- 
tively elementary  concepts.  These  should  be  fully  grasped 
before  their  varied  application  can  be  intelligently  handled. 

33.  To  draw  a  line  containing  a  point  in  each  of  two 
given  dihedral  angles:— Locate  the  points  in  the  specified 
angles  and  draw  the  projections  of  the  line  to  contain  the 
respective  projections  of  the  points. 

A  line  can  also  be  made  to  pass  through  any  one  of 
the  dihedral  angles  severally,  by  a  consideration  of  its 
traces  and  their  proper  location.  For  example,  if  a  line 
goes  from  left  to  right,  and  from  the  1st  angle  through  the 
2nd  into  the  3rd,  its  V  trace  must  be  above  the  G.L.,  and 
its  H  projection  must  touch  the  G.L.  to  the  left  of  the  V 
projection;  its  H  trace  must  be  back  of  the  G.L.,  i.e.,  the 
perpendicular  from  the  Y  projection  of  the  H  trace  must 
meet  the  H  projection  of  the  line  back  of  the  G.L.  If  a 
line  having  the  same  direction  goes  from  the  1st  angle 
through  the  4th  into  the  3rd,  then  the  H  trace  must  be  in 
front  of  the  G.L.,  i.e.,  the  V  projection  must  intersect  the 
G.L.  to  the  left  of  the  H  projection,  and  the  perpendicular 
from  the  H  projection  of  the  V  trace  must  intersect  the  V 
projection  of  the  line  below  the  G.L. 

A  line  lying  in  four  dihedral  angles  is  a  line  lying  in 
either  coordinate  plane  and  crossing  the  G.L.  in  a  finite 
point. 


26  DESCRIPTIVE     GEOMETRY 

34.  The  alphabet  of  the  line  is  an  expression  used  to 
designate  the  possible  positions  of  a  line  with  respect  to 
the  coordinate  planes.  The  directions  of  a  line  are  seven 
in  number. 

1.  Inclined  to  H  and  V. 

2.  Parallel  to  H  and  inclined  to  V. 

3.  Parallel  to  V  and  inclined  to  H. 

4.  Parallel  to  H  and  V  or  parallel  to  the  G.L. 

5.  Perpendicular  or  normal  to  Y. 

6.  Perpendicular  to  H. 

7.  Inclined  to  H  and  V  and  perpendicular  to  the  G.L. 
With  direction  (1)  a  line  may  go  (a)  through  the  1st, 

4th  and  3rd  angles,  (b)  through  the  1st,  2nd  and  3rd 
angles,  (c)  through  the  4th,  1st  and  2nd  angles,  (d) 
through  the  4th,  3rd  and  2nd  angles,  (e)  through  the  G.L. 
and  from  the  1st  to  the  3rd  angles,  or  (f)  through  the 
G.L.  from  the  2nd  to  the  4th  angles. 

With  direction  (2)  a  line  may  go  (a)  through  the  1st 
and  2nd  angles,  (b)  through  the  4th  and  3rd  angles,  or 
(c)  it  may  lie  in  H  and  hence  cut  the  G.L. 

With  direction  (3)  a  line  may  go  (a)  through  the  1st 
and  4th  angles,  (b)  through  the  2nd  and  3rd  angles,  or  (c) 
it  may  lie  in  Y  and  hence  cut  the  G.L. 

With  direction  (4)  a  line  may  lie  (a)  in  the  first  angle, 

(b)  in  the  2nd  angle,  (c)  in  the  3rd  angle,  (d)  in  the  4th 
angle. 

With  direction  (5)  a  line  may  go  (a)  through  the  1st 
and  2nd  angles,     (b)  through  the  4th  and  3rd  angles,  or 

(c)  it  may  lie  in  H  and  go  through  the  G.L. 

With  direction  (6),  a  line  may  go  (a)  through  the  1st 
and  4th  angles,  (b)  through  the  2nd  and  3rd  angles,  or  (c) 
it  may  lie  in  Y  and  go  through  the  G.L. 


THE    LINE    AND    PLANE 


27 


.    .     S' 

i              !     ii 

!  ^ 

i 
1 

1 

j 

a        j    / 

V'            \ 

o^ 

L 

1     L 

T 

Figure    18. 

With  direction  7,  a  line  may  go  (a)  through  the  1st,  *4th 
and  3rd  angles,  (b)  through  the  1st,  2nd  and  3rd  angles, 
(c)  through  the  4th,  1st  and  2nd  angles,  (d)  through  the 
4th,  3rd  and  2nd  angles,  (e)  through  the  G.L.,  the  1st  and 
3rd  angles,  or  (f)  through  the  G.L.,  the  2nd  to  the  4th 
angles. 

The  student  is  advised  to  draw  out  these  positions  for 
himself  as  a  drill  in  determining  traces  and  as  a  reference 
chart  for  the  earlier  work  in  the  subject 


35.    A  point  is  said  to  revolve  about  a  line  as  an  axis 

when  the  path  of  the  point  is  a  circle  whose  plane  is  per- 
pendicular to  the  axis  and  whose  center  is  in  the  axis. 
A  simple  case  is  that  of  the  revolution  of  a  point  about 
a  line  which  is  perpendicular  to  either  coordinate  plane. 
If  the  line  or  axis  of  revolution  is  perpendicular  to  H,  the 
conditions  are  as  shown  in  Fig.  13.  A  point  A  revolves 
about  the  line  O  P.  The  path  of  the  point  being  a  plane 
perpendicular  to  OP,  which  in  turn  is  perpendicular  to  H, 
must  be  parallel  to  H,  hence  has  its  center  at  0  and  is  a 
true  circle  in  H  projection.    Its  vertical  projection  is  a 


28 


DESCBIPTIVE    GEOMETRY 


o' 


G.        Q'         a's        o'         ai        P'      L 


Figure    14. 


limited  straight  line  perpendicular  to  OP  or  parallel  to  the 
G.  L.  with  center  at  o' .  If  the  point  A  moves  through 
any  given  angle,  the  angle  in  its  true  value  is  laid  off  with 
0  as  a  center  and  the  vertical  projection  of  the  point  is 
found  upon  the  vertical  projection  of  the  path.  Let  the 
point  move  toward  V  from  the  position  A  until  it  lies  in 
the  V  plane,  ^i  is  its  H  projection,  where  the  H  projection 
of  the  path  of  the  revolution  crosses  the  G.L.  and  ^i  'is  the 
vertical  projection  upon  the  vertical  projection  of  the  path. 
Another  illustration  is,  when  the  point  revolves  about 
any  line  lying  in  either  coordinate  plane.  Let  A  in  Fig.  14 
be  such  a  point  and  QP  the  given  line.  The  path  of  the 
point,  being  perpendicular  to  the  axis,  is  projected  on  H 
in  this  illustration  as  a  limited  line  perpendicular  to  o  p 
for  OP  lies  in  H  and  a  plane  perpendicular  to  H  will  be 
projected  on  H  in  a  straight  line.  The  center  O  of  the  path 
is  horizontally  projected  where  the  H  projection  of  the 
path  cuts  the  H  projection  of  the  axis.    The  radius  of  re- 


THE   LINE  AND   PLANE  29 

volution  is  the  perpendicular  distance  of  the  point  from  the 
axis.  This  perpendicular  is  oblique,  in  this  case,  to  both 
H  and  V.  To  state  it  in  general  terms  it  is  the  hypotenuse 
of  a  right  angled  triangle  of  which  one  side  is  the  perpen- 
dicular distance  of  the  projection  of  the  point  on  the  plane  of 
the  axis  from  the  axis  and  whose  other  side  is  the  distance 
of  the  point  from  the  plane  of  the  axis.  In  the  example 
under  consideration  the  first  side  mentioned  is  the  distance 
oa,  and  the  second  side  is  the  distance  of  a'  from  the  G.  L. 
This  radius  may  be  derived  by  auxiliary  construction  but 
it  is  better  to  get  it  in  the  following  way:  Conceive  the 
plane  of  the  path  of  the  point  to  be  revolved  about  its  H 
projection  into  H  that  is  about  the  line  oa,  A  falls  at  a^  ; 
a  ai  is  the  second  side  of  the  triangle  just  mentioned  and 
the  revolved  position  of  the  path  is  a  circle  with  oa^  as  a 
radius  and  diameter  ^2  ^3  •  If  A  moves  from  the  given 
position,  in  the  direction  of  the  arrow  in  revolved  position, 
it  falls  in  the  H  plane  at  0^2 »  a  distance  from  0  equal  to 
0  ai  .  If  it  continue  motion  it  will  go  through  the  4th  angle 
again  falling  into  H  at  ^3 . 

Every  point  in  the  path  of  A  shown  in  the  figure  as  re- 
volved into  H  is  horizontally  projected  in  the  diameter 
^2  ag  and  vertically  projected  on  perpendiculars  through 
these  points  as  shown  in  the  figure.  The  lengths  of  the 
ordinates  to  the  diameter  ^2  ^3  are  the  distances  respect- 
ively that  the  points  lie  above  or  below  H  according  to  the 
direction  of  revolution  of  the  path  into  coincidence  with  H. 
In  this  case  all  the  points  in  the  circle  to  the  right  of  the 
diameter  ^2  ^3  stand  for  positions  above  H  and  those  to 
the  left  for  positions  below  H. 

If  it  is  desired  to  revolve  the  point  A  through  a  given 


30 


DESCRIPTIVE     GEOMETRY 


angle  about  the  line  QP  the  angle  can  be  laid  off  with  o 
as  a  center  and  subtended  by  an  arc  of  the  circle  ^3  a^  a^ . 
The  new  position  of  the  point  would  be  horizontally  pro- 
jected by  an  ordinate  to  the  axis  and  vertically  by  laying 
off  the  length  of  this  ordinate  above  or  below  the  Gr.  L.  as 
the  case  may  be  on  a  perpendicular  to  the  G.  L.  through 
the  H  projection. 


Let  A,  Fig.  15,  be  a  point  in  space  to  move  about  a  line 
PQ  in  V,  in  the  direction  of  the  arrow  shown  in  revolved 
position.  The  radius  of  revolution  is  the  hypotenuse 
0  ai'  and  the  revolved  position  of  the  path  is  shown 
at  a\  h' I  d\  ,  A  is  in  the  1st  angle;  if  it  moves  to  the 
position  hi'  shown  as  revolved  into  V  the  V  projection  of 
the  point  is  h'  and  the  H  projection  at  b  is  a  distance  below 
h'  equal  to  the  length  of  the  ordinate  hi'  h' ,  The  point 
is  in  H  in  front  of  V.  If  the  point  moves  from  the  position 
B  to  the  position  C  it  is  in  the  4th  and  3rd  angles.  C  is  a 
point  in  V  below  H  and  hence  c,  in  the  G.  L.,  is  its  H  pro- 
jection. If  the  point  moves  from  the  position  C  to  the  posi- 
tion D,  it  lies  in  the  3rd  and  2nd  angles,    di'  \s>  the  re- 


THE    LINE    AND    PLANE  31 

volved  position  of  the  point  lying  in  H  back  oiY,  d'  is  its 
V  projection  and  d  is  a  distance  from  the  G.  L.  equal  to 
the  length  of  the  ordinate  (^'c^i'.  From  D  to  E  the  point  is 
in  the  2nd  angle.  At  E  the  point  has  come  into  V,  hence 
its  H  projection  is  in  the  G.  L. 

The  student  is  advised  to  copy  the  figure  upon  a  some- 
what larger  scale  and  locate  the  projections  of  a  few 
intermediate  points  in  the  revolution  to  familiarize  himself 
with  the  principles  involved. 

36.  To  revolve  a  point  about  a  line  which  is  oblique  to 
both  coordinate  planes  is  a  much  more  difficult  problem 
than  the  foregoing..  If  it  should  occur  in  any  practical 
problem,  a  good  solution  is  to  change  the  coordinate 
planes;  project  the  axis  and  point  upon  a  Vi  plane  parallel 
to  the  axis,  or  to  project  them  upon  a  plane  through  the 
axis,  and  then  proceed  in  the  manner  outlined. 

37.  A  change  of  one  coordinate  plane  for  a  line  may  be 

made  as  follows:  A  line  may  be  projected  upon  any  new 
Yi  plane  (see  figure  16)  by  finding  the  new  Vi  projections 
of  points  of  the  line.  If  the  G  i  .L  ^ .  is  taken  perpendicular 
to  the  H  projection  of  the  line  obviously  the  Vi  projections 
of  all  points  of  the  line  will  be  in  a  common  perpendicular 
to  the  Gi.Li.,  and  we  have  indeterminate  projections  of 
the  line  as  we  have  when  the  line  lies  in  an  end  plane. 

Project  the  line  upon  a  new  vertical  plane  or  Vg  (see 
Figure  16)  to  distinguish  it  from  the  one  just  mentioned, 
whose  G2 .  L  2 .  is  parallel  to  the  H  projection  of  AB.  Obvi- 
ously V2  is  parallel  to  the  line  since  G2 .  Lg.  is  parallel  to 
ah  which  is  the  H  trace  of  the  H  projecting  plane  of  AB. 
Finding  the  projection  of  AB  on  Va  in  al  h\,  we  have  a 


32 


DESCRIPTIVE     GEOMETRY 


Figure  16. 

projection  which  is  parallel  to  the  line  and  of  the  same 
length  as  the  line.  This  shows  one  of  the  uses  of  a  change 
of  coordinate  plane. 

To  project  a  line  upon  a  new  Yi  plane  whose  Gi .  L^ . 
is  parallel  to  the  H  projection  of  the  line  results  projec- 
tively  in  the  same  way  as  considering  the  H  projecting 
plane  of  the  line.  And  since  the  Vi  projection  is  revolved 
about  the  G-i.  Lj.  into  coincidence  with  H,  and  further 
since  a  revolution  about  the  Y  projection  into  Y  is  sim- 
ilar to  a  revolution  about  the  H  projection  into  H,  we 
may  state  as  the  analysis  for  finding  the  true  length  of  a 
line  by  projection: 

Analysis:— Bevolve  the  line  about  either  projection  into 
the  coordinate  plane  of  that  projection.  The  revolved 
position  is  the  true  length  required. 


38.    To  find  the  true  length  of  a  line  by  revolution. 

Ansiiysis:— Revolve  the  line  about  the  projecting  line  on 
V  or  H  of  any  one  of  its  points  until  the  line  is  parallel  to  H 
or  V,  Its  projection  on  the  plane  to  which  it  is  parallel, 
will  give  the  true  length  required. 


THE    LINE    AND    PLANE 


33 


FlQUBK    17     AND     18. 

Construction: — A  convenient  point  is  one  extremity  of 
the  limited  line.    See  Figure  17. 

Revolve  the  line  AB  about  the  H  projecting  line  a' o' 
of  the  point  A  on  H*  until  a&  is  parallel  to  the  G.L.  The 
point  h  travels  in  the  arc  of  a  circle  with  a  as  a  center. 
This  circle  being  parallel  to  H,  is  projected  on  V  as  a 
straight  line  parallel  to  the  G.L.,  whence  })\  the  inter- 
section of  such  a  parallel  with  a  perpendicular  to  the  G.L. 
through  6i  is  the  position  of  the  point  in  vertical  projection 
when  the  line  is  parallel  to  V.  a'6'i  is  then  the  true 
length  of  the  line. 

Similarly,  the  line  might  be  revolved  about  the  V  pro- 
jecting line  of  either  A  or  B  until  the  line  AB  is  parallel 
to  H,  whence,  in  its  revolved  position,  the  H  projection 
would  show  the  true  length  of  the  line.     (See  Figure  18.) 


*a'o  Is  the  vertical  projection  of  the  H  projecting  line. 


CHAPTER    III. 


PROBLEMS  IN  POINT,  LINE  AND  PLANE. 


39.  THEOREM  IV. 

If  a  line  lies  in  a  plane  the  traces  of  the  line  lie  in  the 
corresponding  traces  of  the  plane. 

Proof:— By  geometry,  a  line  lies  in  a  plane  when 
every  point  of  the  line  lies  in  the  plane.  The  trace  of  a 
plane  with  a  coordinate  plane  is  a  line  lying  wholly 
within  the  plane.  It  must  intersect  every  other  line 
of  the  plane,  each  in  some  point.  Such  a  point  being  on 
another  line  of  the  plane,  must  be  the  trace  of  that  line 
with  the  coordinate  plane. 

The  converse  of  this  is  not  always  true,  for  example,  if 
a  line  goes  through  the  G.L.  at  the  point  of  intersection 
with  the  G.L.  of  any  plane,  the  traces  of  the  line  will 
coincide  with  each  other  and  lie  upon  the  traces  of  the 
plane  but  the  line  will  not  necessarily  lie  in  the  plane 
unless  some  other  point  also  does. 

40.  THEOREM  V. 

A  given  line  is  parallel  to  a  plane  when  it  is  parallel  to  a 
line  in  the  plane  or  when  it  lies  in  a  parallel  plane. 


PROBLEMS    IN     POINT,    LINE     AND    PLANE  35 

Proof:— Since  from  geometry  parallels  between  par- 
allels are  equal,  and  conversely,  if  from  any  two  points 
in  the  line  parallels  can  be  drawn  which  also  terminate 
in  the  given  plane  and  are  of  equal  length,  they  will 
pierce  the  given  plane  in  points  which  lie  on  a  line 
parallel  to  the  given  line.  Hence  the  given  line  is  par- 
allel to  the  plane.  And  since  what  is  true  of  one  line 
will  also  be  true  of  either  line,  a  line  is  parallel  to  a 
plane  when  it  lies  in  a  parallel  plane. 

41.  Corollary.  If  a  line  is  parallel  to  either  coordinate 
plane,  then  any  plane  passed  through  the  line  will  have 
its  trace  on  that  coordinate  plane  parallel  to  the  projection 
of  the  line  on  that  plane. 

42.  Every  point  in  the  V  trace  of  a  plane  is  a  point  in  V 

as  well  as  a  point  in  the  plane,  hence  the  corresponding 
projection  is  in  the  G.L.  Similarly  every  point  in  the  H 
trace  of  a  plane  is  a  point  in  H  as  well  as  a  point  in  the 
plane,  hence  the  corresponding  or  Y  projection  of  this 
point  is  also  in  the  G.L.  To  locate  any  other  points  lying 
in  the  plane  conceive  of  a  line  or  lines  lying  in  it,  passing 
through  the  point,  and  limited  by  the  traces  of  the  plane. 

43.  To  pass  a  plane  through  three  given  points. 

Analysis:— Since  the  points  lie  in  a  plane  they  will  lie 
upon  lines  of  the  plane  passing  through  them.  Hence  if 
lines  are  drawn  connecting  the  points  the  traces  of  two  such 
lines  are  sufficient  to  locate  the  traces  of  the  plane  of  the 
lines,  that  is  the  plane  of  the  points. 


36 


DESCRIPTIVE     GEOMETRY 


Q. 


^ 


/   I     I, 

r/       \u'    1  IS 


t^^i:^ 


\Q 


-^ 


/ 


/ 


yaf^ 


/      -^-' 


\ 


I  / 


/ 


/        x/ 


Figure  19. 

Construction:— (See  Fig.  19). 

Let  A,  B  and  C  be  three  points  in  projection.  Connect 
A  and  B  by  a  line,  also  A  and  C.  Q  is  found  to  be  the  V 
trace  of  AB  (by  Sec.  28)  and  E  the  V  trace  of  AC.  The  V 
trace  of  the  plane  of  the  three  points  passes  through  the 
points  r'  and  q' ,  Similarly  U  is  the  H  trace  of  the  line 
AB  and  S  is  the  H  trace  of  the  line  AC,  and  the  H  trace  of 
the  plane  of  the  three  points  goes  through  the  points  u  s. 
If  the  work  is  done  accurately  the  H  and  V  traces  will 
meet  on  the  G.L.  Hence,  if  one  trace  is  obtained  by  the 
method  described  only  one  point  is  required  in  the  other 
trace. 

If  one  of  the  lines  drawn  connecting  two  of  the  three 
points  is  parallel  to  the   G,L,  then    any   plane    passed 


PROBLEMS    IN    POINT,    LINE    AND    PLANE  37 

through  this  line  is  parallel  to  the  G.L.  by  Theorem  V, 
Sec.  41,  and  the  traces  of  the  plane  are  parallel  to  the  G.L. 
Hence,  it  is  only  necessary  to  locate  one  point  in  each 
trace. 

If  the  traces  of  lines  connecting  three  points  in  space  do 
not  come  within  available  limits  on  the  drawing,  then  any 
auxiliary  lines  may  be  taken  to  intersect  those  connecting 
the  three  points.  These  can  be  so  chosen  that  their  traces 
can  be  readily  obtained. 

If  the  lines  connecting  the  projections  of  the  points  are 
perpendicular  to  the  G.L.,  it  is  necessary  to  project  the 
points  upon  an  end  plane;  if  one  line  connecting  two 
points  is  perpendicular  to  the  G.L.,  then  an  auxiliary  line 
may  be  used. 

The  problem  to  pass  a  plane  through  a  line  and  a  point 
does  not  differ  from  the  preceding.  Draw  an  auxiliary 
line  through  the  point  to  intersect  the  given  line  and 
proceed  as  before.  A  convenient  auxiliary  line  to  use  is 
one  which  is  parallel  to  the  given  line. 

44.  To   draw  the  traces  of  a  plane  which  shall  contain 
a  given  line. 

Analysis: — Since  an  infinite  number  of  planes  can  be 
passed  through  a  line,  draw,  through  the  V  and  H  traces 
of  the  line,  the  V  and  H  traces  of  any  plane  which  will 
intersect  each  other  in  the  G,L, 

45.  The    horizontal    and    the  vertical  of  a    plane   are 

respective  lines  in  the  plane,  the  first  of  which  is  parallel 
to  H  and  the  second  is  parallel  to  Y. 


38 


DESCRIPTIVE    GEOMETRY 


Figure    20. 

For  example,  see  Figure  20,  the  line  AB  is  a  horizontal 
of  the  plane  T  for  its  H  projection  is  parallel  to  the  H 
trace  of  the  plane  T,  and  its  V  projection  is  parallel  to  the 
G.L.  And  similarly,  CD  is  a  vertical  of  the  plane  T  for  its 
V  projection  is  parallel  to  the  Y  trace  of  the  plane  and  its 
H  projection  is  parallel  to  the  G.L.  These  lines  are 
useful  in  the  solution  of  problems. 

It  is  useful,  sometimes,  to  consider  a  line  as  the 
intersection  of  two  planes,  for  example,  the  line  AB  of 
Figure  20  is  the  line  of  intersection  of  the  plane  T  with 
a  Vi  plane  which  is  parallel  to  the  H  trace  of  T  and  whose 
Gi.  Li.  coincides  with  ah.  Its  vertical  trace  will  be  the 
line  W .  In  like  manner  the  line  CD  is  the  line  of  inter- 
section of  the  plane  T  with  a  new  Hi  plane  which  is 
parallel  to  the  V  trace  of  T  and  whose  G-g.  L2.  coincides 
with  the  line  c' d' ,    Its  H  trace  will  be  the  line  d' d. 


46.    To  assume  a  point  in  a  plane. 
Analysis: — Since  lines  lying  in  the  plane  and  passing 


PROBLEMS    IN    POINT,   LINE    AND    PLANE  39 

through  the  point  have  their  traces  in  the  respective  traces 
of  the  plane,  one  projection  of  such  a  line  may  he  assumed 
at  will  to  contain  one  assumed  projection  of  the  point,  and 
hy  means  of  the  traces  of  the  line,  the  corresponding 
projection  may  he  found,  which  in  turn  contains  the  corres- 
ponding projection  of  the  point. 

The  horizontal  or  vertical  of  the  plane  are  convenient 
lines  to  use  for  this  purpose.  E  is  a  point  in  the  plane  T, 
Figure  20,  found  by  the  method  described. 

47.  THEOREM  VI. 

The  traces  of  parallel  planes  with  a  third  plane  are 
parallel. 

Proof:  By  hypothesis  the  traces  each  lie  in  the  re- 
spective planes  and  also  in  a  third  plane  together.  By 
Theorm  V,  if  through  any  two  points  of  one  of  the  lines 
or  traces  parallels  can  be  drawn,  lying  in  the  same 
plane,  and  of  the  same  length  the  line  connecting  their 
extremities  will  be  parallel  to  the  line  or  trace.  But 
the  two  such  lines  which  may  be  drawn  to  connect  the 
points  on  the  second  trace  are  themselves  in  a  third 
plane  together  with  the  first  line,  and  the  line  connecting 
their  extremities  lies  in  the  third  plane  and  is  parallel 
to  the  first  line  or  trace. 

48.  Both  coordinate  planes  may  be  changed  with  respect 
to  a  line.  Let  AB,  Fig.  21,  be  the  line  to  project  upon  any 
new  Vi  plane  and  Hi  plane.  Find  the  Vi  projection  by 
Sec.  24.  Let  the  new  Hi  plane  be  chosen  by  its  G2  .Lg ,  it 
is  perpendicular  to  the  V,  plane.    The  points  A  and  B  are 


40 


DESCRIPTIVE    GEOMETRY 


Figure  21. 

projected  upon  it  by  drawing  perpendiculars  from  their 
Vi  projections  to  the  G2-L2-  The  distances  of  the  points 
from  Vi  are  fixed  and  their  projections  upon  any  Hi  plane 
are  as  far  from  the  ground  line  of  that  plane  with  Y^  as 
the  H  projections  are  from  the  ground  line  with  Vi  hence 
^2  and  h^are  on  perpendiculars  to  G-g  .L2 .  through  a\  and 
h\  and  distant  respectively  from  G2.L2.  the  amounts  that 
a  and  h  are  from  the  G 1  .L  i . 


49.    Change  of  one  coordinate  plane  with  respect  to  any 
oblique  plane:    Let  T  be  any  plane,  see  Fig.  22.    Choose 

/  / 


F  IGUBE    22. 


any  new  Vi  plane  with  Gi  .Li .  cutting  TH  in  the  point  A. 
The  plane  Vi  cuts  a  line  from  V  which  is  perpendicular  to 


PROBLEMS    IN     POINT,    LINE     AND     PLANE 


41 


the  G.  L.  because  Vi  is  perpendicular  to  H,  this  line  is  hb' . 
h'  is  also  a  point  in  T  because  it  is  in  the  V  trace  of  T. 
The  Vi  projection  of  the  point  B  is  on  a  perpendicular  to 
the  Gi.Li.  through  &,  namly  &'i  .  The  Vi  projection  of  A 
is  a  because  the  point  lies  at  the  intersection  of  TH  and 
Gi  .Li .  It  must  moreover  be  a  point  in  the  new  Vi  trace 
of  the  plane  T.  Hence  ab\  is  the  Vj  trace  of  the  plane  T. 
If  Gi.Li,  had  been  taken  perpendicular  to  TH,  by  in- 
spection, we  can  see  that  the  plane  T  would  have  been  per- 
pendicular to  Vi  . 

/    / 


Figure    23. 

50.  Change  of  both  coordinate  planes  with  respect  to  a 
plane:  Let  T  be  any  plane,  Fig.  23.  Find  its  V,  trace 
as  in  the  preceding  paragraph.  Now  take  any  new 
Hi  plane  cutting  V,  in  the  line  G^.L^.  Where  TV^  cuts 
G2.L0.  is  one  point  in  the  H^  trace  of  the  plane,  i.  e.,  C.  The 
plane  Hx  cuts  a  line  from  H  which  is  perpendicular  to  V^^ , 
because  both  planes  are  perpendicular  to  V;  this  line  is 
DE.  E  is  also  a  point  in  T  because  it  is  in  the  H  trace  of 
T.    The  H  projection  of  E  is  on  a  perpendicular  to  Gi  .  Li . 


42 


DESCRIPTIVE     GEOMETRY 


through  e'.  The  Hi  projection  of  E  is  on  a  perpendicular 
to  G2  .Ls .,  a  distance  from  it  equal  to  d  e.  It  is  a  point 
in  the  new  Hi  trace  of  the  plane  T.  C  is  another  point  in 
the  H  trace.  Hence,  the  line  THi  connecting  C2  and  e^  will 
be  the  Hi  trace  of  the  required  plane. 


\     ^^ 


Figure    24. 


Consider  another  case  of  change  of  coordinate  planes, 
with  respect  to  any  oblique  plane.  Let  T,  Fig.  24,  be  the 
given  plane  and  let  it  be  required  to  take  a  new  Vj  plane 
perpendicular  to  it  and  a  new  Hi  plane  parallel  to  it.  The 
Gi  Li  of  Vi  is  perpendicular  to  TH,  for  if  a  plane  is  per- 
pendicular to  y  its  H  trace  will  be  perpendicular  to  the 
G.  L.  By  preceding  paragraph  we  find  TVi  .  The  new 
Hi  plane,  to  be  parallel  to  T  must  have  its  G^.La.  parallel  to 
the  Vi  trace  of  the  plane  or  TV  i .  The  plane  Hi  cuts  a  line 
from  H  which  is  perpendicular  to  Vi  but  this  perpendicular 
by  construction  will  be  parallel  to  TH,  meeting  it  at  in- 
finity. Hence  since  TVi  is  parallel  to  G2  L2 ,  THi  will  also 
be  parallel  to  and  at  an  infinite  distance  from  G2  .L2 . 


PROBLEMS    IN    POINT,    LINE     AND    PLANE 


43 


51.    To  draw  a  line  parallel  to  a  given  plane  through  a 
given  point 

Analysis:— From  Theorem  F,  a  line  is  parallel  to  a  plane 
when  it  is  parallel  to  a  line  lying  in  the  plane.  There  can 
be  an  infinite  number  of  lines  through  a  point  parallel  to  a 
plane,  each  will  be  parallel  to  a  line  in  the  plane,  hence 
draw  any  line  lying  in  the  given  plane  and  through  the 
point  draw  the  projections  of  a  line  parallel  to  it. 


Figure  25. 


Construction:— Let  T,  Figure  25,  be  an  oblique  plane 
and  C  the  given  point.  Draw  any  line  AB  in  T  (whose 
traces  will  lie  in  the  traces  of  the  plane).  Through 
c  draw  a  line  parallel  to  ab  and  through  c '  a  line  parallel 
\jQ>  a'b' ,    The  line  through  C  is  parallel  to  the  plane  T. 

Since  an  infinite  number  of  lines  can  be  drawn  through 
C  and  parallel  to  T,  they  will  constitute  a  plane,  so  if  two 
lines  are  drawn  through  C  parallel  respectively,  to  two 
lines  lying  in  T,  they  will  lie  in  a  plane  parallel  to  T  through 
the  given  point.  But  such  a  plane,  by  theorem  VI,  would 
also  have  its  traces  parallel  to  those  of  T,  hence  it  is  not 
necessary  to  draw  but  one  line  throuiorh  C  to  locate  it. 


44 


DESCRIPTIVE    GEOMETRY 


52.    To  pass  a  plane   through   one  line  and   parallel  to 
another  line. 

Analysis: — By  Theorem  F,  a  line  is  parallel  to  a  plane 
when  it  is  parallel  to  a  line  lying  in  the  plane.  Hence  draw 
through  a  point  in  the  first  line,  a  line  parallel  to  the 
second.  The  plane  of  the  two  intersecting  lines  will  he 
parallel  to  the  second  line. 


Figure  2( 


Construction:— In  Figure  26,  let  AB  be  a  line  through, 
which  to  pass  a  plane  parallel  to  the  line  CD.  Through 
any  point  of  AB  as  O  draw  an  auxiliary  line  EF  parallel 
to  CD.    The  plane  of  AB  and  EF  is  the  plane  required. 

If  the  two  given  lines  are  parallel  to  each  other,  i.e., 
their  projections  respectively  parallel  then  any  plane 
through  the  one  will  be  parallel  to  the  other. 

If  either  one  of  the  two  given  lines  is  parallel  to  the  G, 
L.,  the  required  plane  will  be  parallel  to  the  G.L.  and  its 
traces  will  be  parallel  to  the  G.L. 

If  either  one  or  both  of  the  lines  have  their  projections 
perpendicular  to  the  G,L,,  then  the  traces  are  obtained  by 
use  of  an  end  plane. 


/ 


PROBLEMS    IN    POINT,   LINE    AND    PLANE 


45 


There  can  be  but  one  plane  containing  a  given  line 
and  parallel  to  another  line  unless  the  two  lines  are 
parallel  to  each  other. 


53,  Within  any  given  plane  perpendicular  to  a  coordi- 
nate plane,  to  draw  a  line  making  a  given  angle  with  that 
coordinate  plane.  \    / 

Analysis:— j5^  TJieorem  II,  one  projection  of  the  line 
will  coincide  with  the  trace  of  the  plane,  that  projection  on 
the  coordinate  plane  to  which  the  given  plane  is  perpen- 
dicular; hence  we  can  assume  that  projection  at  will, 
Bevolve  the  assumed  projection  together  with  the  trace  of 
the  given  plane  into  the  corresponding  coordinate  plane 
about  the  corresponding  trace  of  the  plane  as  an  axis.  The 
angle  can  here  be  constructed  its  true  value.  Bevolve  the 
line  back  again  to  its  original  position,  and  both  projections 
of  the  line  and  the  angle  will  be  shown. 

Construction:  —  Let  Fig.  27,  show  a  plane  perpen- 
dicular to  V.  Required  a  line  AB  lying  in  it  and  making  an 
angle  of  60'  with  V.    Assume  a'b'  with  A  a  point  in  V. 


46  DESCRIPTIVE     GEOMETRY 

Revolve  AB  about  TH  as  an  axis  until  A  fall  into  H.  The 
angle  at  a^  with  the  the  G.L.  as  one  side  will  be  the  true 
angle  and  h^  can  be  obtained  in  the  manner  shown. 
Eevolve  the  line  back  again  into  its  original  position,  a^ 
moves  along  the  G.L.  as  in  the  figure  to  a  at  the  inter- 
section with  a  perpendicular  from  a'  to  the  G.L.  and  h^ 
moves  parallel  to  the  G.L.  to  the  position  &  on  a  perpen- 
dicular to  the  G.L.  through  h' ,  whence  ah  is  the  H 
projection  of  the  line. 

If  the  plane  T  has  hut  one  trace,  which  would  of  course 
be  parallel  to  the  G.L.  then  one  projection  of  the  required 
angle  would  coincide  with  the  given  trace  of  the  plane  and 
the  other  projection  of  the  angle  would  at  once  show  its 
true  value. 

If  the  given  plane  is  perpendicular  to  hoth  coordinate 
planes  the  angle  is  obtained  by  revolving  the  given  plane 
into  Y  about  its  V  trace.  The  projection  of  the  angle  on 
the  CT'd  plane  would  show  its  true  value. 

54.    Through  a  given  point  to  pass  a  plane  parallel  to  two 
giv<?n  lines. 

Analysis:—^  plane  is  parallel  to  a  line  when  it  contains 
a  line  parallel  to  the  given  line.  Therefore  through  the 
given  point  draw  lines  parallel  respectively  to  the  given 
lines,  the  plane  of  these  lines  is  the  required  plane. 

Construction:  Let  O  Fig.  28  be  the  given  point,  AB 
and  CD  respectively,  the  given  lines.  Through  O  draw  two 
lines,  parallel  respectively  to  AB  and  CD.  The  V  traces 
of  these  lines  are  respectively  Ai  and  D^  and  the  H  traces, 
Bi  and  Ci  ,  Ai  and  D^  therefore,  are  points  in  the  V  trace 
of  the  plane  of  the  two  lines  and  B^  and  Ci  are  points  in 
the  n  trace,  which  constitute  the  plane  required. 


PROBLEMS    IN    POINT,  LINE    AND     PLANE  47 


/ 


G.    ! 


^^^    di    «i 


Figure  28. 


K 


If  the  given  point  is  in  the  G,  L,  the  traces  of  the  re- 
quired plane  intersect  each  other  in  this  point  and  hence 
after  drawing  the  two  auxiliary  lines  through  the  point, 
draw  a  third  auxiliary  line  lying  in  the  plane  of  the  two, 
i.  e.,  intersecting  them;  the  traces  of  this  line  will  lie  on 
the  traces  of  the  required  plane  thus  giving,  together  with 
the  given  point,  two  points  in  each  trace  of  the  plane. 

If  it  is  found  that  any  third  auxiliary  line  just  mention- 
ed also  has  both  of  its  traces  in  the  G.  L.  then  the  traces  of 
the  required  plane  are  in  the  G.  L. 

If  both  given  lines  are  parallel  to  the  G,  L,  the  required 
plane  will  be  parallel  to  the  G.  L.  It  is  necessary  either  to 
use  a  third  auxiliary  line  as  in  the  case  just  preceding  or 
an  end  plane  whereon  the  traces  of  the  auxiliary  lines  will 
establish  the  trace  of  the  required  plane  with  the  end  plane 
and  from  this  its  traces  with  V  and  H  may  be  obtained. 


55,    To  find  the  line  of  intersection  of  two  planes; 

Analysis: — Since  the  required  line  is  a  line  of  each 
plane  its  traces  will  lie  in  both  planes  respectively,  i.e,  its 


48 


DESCRIPTIVE    GEOMETRY 


V  trace  will  he  at  the  point  of  intersection  of  the  V  traces 
of  the  given  planes  and  its  H  trace  at  the  point  of  inter- 
section of  their  H  traces.  Hence  the  line  connecting  these 
points  will  be  the  line  required. 


Figure  29. 

Construction:— Let  the  two  planes  be  given  as  in  Figure 
29.  The  point  of  intersection  of  the  H  traces  is  A  and 
it  is  the  H  trace  of  the  line  of  intersection.  The  point  of 
intersection  of  the  V  traces  is  B  and  it  is  the  V  trace  of  the 
line  of  intersection,  hence  AB  is  the  line  required. 

If  the  traces  of  both  planes  do  not  intersect  at  accessible 
points  upon  the  drawing,  auxiliary  construction  must  be 
resorted  to  as  follows : 


56<  To  find  the  line  of  intersection  of  two  planes  when 
one  or  both  pairs  of  traces  do  not  intersect  at  accessible 
points. 

Analysis:— J.^2/  plane  parallel  to  a  coordinate  plane 
will  cut  from  each  plane  a  line  parallel  to  that  coordinate 
plane.  These  lines,  lying  in  each  plane  must  intersect 
upon  the  line  of  intersection  of  the  two  given  planes. 
Hence,  pass  one  or  more  auxiliary  planes  parallel  to  H  or 
V  and  find  two  pairs  of  lines  cut  from  each  plane.     The 


PROBLEMS    IN    POINT,   LINE    AND    PLANE 


49 


points   of  intersection  of  these  lines  will   determine    the 
required  line. 

Construction:— Let  the  planes  T  and  S  be  given  as  in 


FlQUBK    80. 


Figure  30,  neither  pair  of  traces  intersecting  at  accessible 
points.  Pass  any  two  auxiliary  planes  P  and  E  parallel  to 
V.  They  will  cut  lines  AB  and  CB  respectively  from  the 
planes  T  and  S,  intersecting  each  other  in  the  point  B  and 
lines  DE  and  FE  respectively!  from  the  planes  T  and  S 
intersecting  each  other  in  the  point  E.  The  line  BE  is  the 
required  line  of  intersection  of  the  two  planes. 

If  only  one  pair  of  traces  intersect  at  an  inaccessible 
point,  the  following  convenient  construction  may  be 
employed: 

Analysis:— Para^^e?  planes  intersect  a  third  plane  in 
parallel  lines,  hence  draw  an  auxiliary  plane  parallel  to 
one  of  the  given  planes  and  it  will  intersect  the  other 
given  plane  in  a  line  tvhich  is  parallel  to  the  required  line. 


50 


DESCRIPTIVE    GEOMETRY 


FlQUKB    81. 


Construction: — Let  the  planes  S  and  T  be  given  as  in 
Figure  31.  At  any  point  upon  the  G.L.  draw  a  plane  Ti 
parallel  to  T,  its  traces  will  be  parallel  to  T  and  by  See.  47, 
A 1 B 1  will  be  its  line  of  intersection  with  S.  Now  a'  and  a 
are  the  projections  of  the  V  trace  of  the  line  of  inter- 
section of  S  and  T,  hence  through  A  draw  the  projections 
of  aline  parallel  to  AiBi  and  it  will  be  the  required  line. 

If  both  of  the  given  planes  cut  the  G,L,  at  a  common 
point,  either  of  the  two  preceeding  special  constructions 
may  be  employed,  noting  that  the  line  of  intersection  will 
pierce  the  G.L.  at  the  same  point  as  the  planes. 

If  both  of  the  given  planes  have  traces  parallel  to  the 
G.L,,  their  line  of  intersection  will  be  parallel  to  the  G.L. 
and  the  intersection  of  the  planes  with  an  end  plane  is 
necessary  to  determine  the  distance  of  this  line  from  the 
coordinate  planes  in  both  projections.  Its  projection  on 
the  end  plane  will  be  the  point  of  intersection  of  the  traces 
of  the  given  planes  with  the  end  plane. 


PROBLEMS    IN    POINT    LINE    AND    PLANE 


51 


When  two  given  planes  are  parallel^  their  line  of  inter- 
section is  at  infinity  and  its  projections  are  at  infinity  in 
both  coordinate  planes. 

WJien  two  planes  have  one  pair  of  parallel  traces,  the 
line  of  intersection  will  have  its  similar  projection  parallel 
to  the  parallel  traces,  the  corresponding  projection  will  be 
parallel  to  the  G.L. 


Figure  82. 


57.    To  find  the  trace  of  a  line  with  any  plane. 

Analysis',— The  required  trace  will  lie  upon  the  line  of 
intersection  with  the  given  plane  of  any  plane  passed 
through  the  given  line,  hence  pass  any  auxiliary  plane 
through  the  line,  find  its  line  of  intersection  with  the  given 
plane  and  where  the  given  line  pierces  this  line  of  intersec- 
tion will  he  the  trace  required.  The  most  convenient 
auxiliary  plane  for  this  purpose  is  either  the  V  or  the  H 
projecting  plane  of  the  line. 

Construction:— Let  T  in  Figure  32  be  any  plane  and 
AB  any  line  oblique  to  the  plane  T,  to  H  and  to  V.    Draw 


52  DESCRIPTIVE     GEOMETRY 

the  H  projecting  plane,  S,  of  the  line  AB.  Find  its  line  of 
intersection,  CD,  with  the  plane  T.  Note  that  the  H  pro- 
jection of  this  line  will  be  coincident  with  the  H  trace  of 
the  plane  S,  since  S  is  perpendicular  to  the  H  plane. 
Where  CD  intersects  the  line  AB  in  E  is  the  trace 
required. 

If  tJie  plane  is  ohlique  and  the  line  is  parallel  to  the 
G.L.,  either  projecting  plane  of  the  line  will  have  one  trace 
at  infinity  and  the  line  of  intersection  of  the  projecting 
plane  and  the  given  plane  will  have  one  projection  parallel 
to  the  similar  trace  of  the  given  plane,  the  corresponding 
projection  will  be  parallel  to  the  Gr.L. 

If  the  given  plane  is  an  end  plane,  the  point  of  inter- 
section of  the  projections  of  the  line  with  the  traces  of  the 
plane  will  be  the  projections  of  the  point  of  intersection  of 
the  line  with  the  plane. 

If  the  given  plane  is  perpendicular  to  one  coordinate 
plane,  and  oblique  to  the  other,  while  the  line  is  oblique  to 
both,  then  note  that  every  point  in  the  given  plane  is 
projected  upon  that  plane  to  which  it  is  perpendicular  in 
its  trace  upon  that  plane.  Hence  one  projection  of  the 
piercing  point  will  be  where  the  one  projection  of  the  line 
cuts  the  trace  of  the  plane  with  the  coordinate  plane  to 
which  the  latter  is  perpendicular. 

If  the  given  plane  is  ohlique  and  the  given  line  is  per- 
pendicular to  the  G.L.,  then  the  projecting  planes  of  the 
line  coincide  and  are  together  an  end  plane.  Find  the 
line  of  intersection  of  the  given  plane  with  the  end  plane 
and  the  piercing  point  will  be  on  this  line. 

58.  THEOREM  VIL 

If  a  line  is  perpendicular  to  a  plane  the  projections  of 


PROBLEMS    IN    POINT,  LINE    AND    PLANE  53 

the  line  are  perpendicular  respectively  to  the  traces  of  the 
plane. 

Proof: — The  projecting  plane  of  a  line,  whose  one 
trace  is  identical  with  the  projection  of  the  line,  is  per- 
pendicular to  the  coordinate  plane,  and  is  also  perpendic- 
ular to  the  given  plane  since  it  contains  a  line  perpendic- 
lar  to  the  given  plane;  hence  it  is  perpendicular  to  the 
line  of  intersection  of  the  two;  but  this  line  of  inter- 
section is  the  trace  of  the  given  plane  with  the  coordinate 
plane,  hence,  either  projection  of  the  line  will  be  per- 
pendicular to  the  similar  trace  of  the  plane. 

59.    Through  a  given  point  to  pass  a  plane  perpendicular 
to  a  given  line. 

Analysis:—^!/  Theorem  VII,  the  traces  of  the  required 
plane  are  perpendicular  respectively  to  the  projections  of  the 
line.  Therefore  through  the  given  point  draw  a  vertical  or 
a  horizontal  of  the  required  plane  since  one  projection  of 
which  will  have  the  same  direction  as  the  similar  trace  of 
the  required  plane.  The  trace  of  this  line  upon  the  cor- 
responding plane  of  projection  will  he  one  point  in  the 
corresponding  trace  of  the  required  plane.  Both  traces  of 
the  required  plane  can  then  be  drawn  since  they  must 
intersect  upon  the  G.L, 

Construction:— Let  AB,  Figure  33,  be  a  given  line  and 
O  the  given  point.  Through  O  draw  the  line  OP  as  a 
horizontal  of  the  required  plane,  i.e.,  op  is  perpendicular 
to  ah  and  o'p'  is  parallel  to  the  G.L.  The  V  ^trace,  C,  of 
the  line  OP  is  a  point  in  the  vertical  trace  of  the  required 
plane  and  the  H  trace  goes  through  the  point  in  which  the 


54 


DESCRIPTIVE    GEOMETRY 


FIGURE    88. 

V  trace  touches  the  Gr.L.,  both  traces  being  perpendicular 
respectively  to  the  projections  of  AB. 

If  the  given  point  is  in  the  G.L,,  the  traces  of  the 
required  plane  will  intersect  each  other  in  this  point. 

If  the  given  line  is  in  an  end  plane  and  the  given  point 
is  in  the  G,L,,  then  the  traces  of  the  required  plane  are  in 
the  G.L.,  and  the  plane  is  indeterminate  in  position.  It 
will  be  completely  determined  by  ascertaining  the  piercing 
point  of  the  given  line  with  it.*  This  is  done  by  project- 
ing both  plane  and  line  upon  an  end  plane  since  the  line 
is  parallel  and  the  plane  is  perpendicular  respectively  to 
the  end  plane.  The  point  in  which  the  end  projection  of 
the  line  cuts  the  end  trace  of  the  plane  is  the  end  projec- 
tion of  the  piercing  point  of  the  line  with  the  plane. 

If  the  given  line  lies  in  an  end  plane  and  the  point  is 
outside  of  the  line  then  the  required  plane  is  parallel  to  the 
G.L.,  and  contains  a  line  through  the  given  point  projec- 


*A  plane  whose  traces  are  lu  the  G.L.   is  completely  determined  by  the 
location  of  a  point  in  the  plane. 


PROBLEMS   IN    POINT,    LINE    AND    PLANE  55 

tively  perpendicular  to  the  given  line.*  Hence  to  find  the 
traces  of  the  plane,  project  the  line  and  point  upon  an  end 
plane.  Draw  through  this  projection  of  the  point  the  end 
projection  of  a  line  perpendicular  to  the  end  projection  of 
the  given  line.  The  traces  of  this  line  lie  upon  the  traces 
of  the  required  plane. 

60.    To  find  the  distance  of  a  point  from  a  plane. 

Analysis:— TAe  distance  of  a  point  from  a  plane  is  the 
length  of  a  perpendicular  from  the  point  to  the  plane. 
Hence  draw  through  the  point  a  line  perpendicular  to  the 
plane.  Find  the  piercing  point  of  this  line  with  the  plane, 
[by  Sec.  57].  Next  find  the  distance  between  the  two 
points  on  the  line  by  any  of  the  methods  previously 
explained. 

Construction; — Let  T,  Figure  34,  be  a  given  plane  and 
O  a  given  point.  Through  O  draw  OR  the  indefinite  per- 
pendicular to  the  traces  of  the  plane  T.  By  means  of  the 
H  projecting  plane  S  of  the  line  OR,  and  its  line  of  inter- 
section with  the  plane  T,  obtain  the  piercing  point  P. 
The  length  of  the  line  connecting  O  and  P  can  then  be 
obtained  by  revolving  the  line  about  O  as  a  center  until  it 
is  parallel  to  H  whence  op^  is  the  length  required. 

If  the  given  plane  is  an  end  plane,  the  perpendicular 
distance  required  is  a  line  parallel  to  both  H  and  V,  and 
hence  is  projected  its  true  length  in  the  line  drawn  through 
the  projection  of  the  point  and  perpendicular  to  the  traces 
of  the  plane. 


*  Projectively  perpendicular  means  that  the   projections  on   the  plane  are 
perpendicular. 


56 


DESCEIPTIVE    GEOMETRY 


/ 


/ 


Figure  84. 

If  the  given  plane  is  parallel  to  the  G,L.,  the  perpen- 
dicular distance  required  is  a  line  lying  in  an  end  plane, 
hence  to  find  its  length  project  both  plane  and  point  upon 
an  end  plane  whence  the  distance  between  the  end  pro- 
jection of  the  point  and  the  end  trace  of  the  plane  is  the 
required  distance. 


61.     Definitions; — 

From  geometry  we  have  by  definition :  — a  line  is  per- 
pendicidar  to  a  plane  ivhen  it  is  perpendicular  to  every  line 
in  the  plane  through  its  foot;  or,  in  other  words,  all 
of  the  perpendiculars  to  a  straight  line  at  a  given  point  in 
it  lie  in  a  plane  perpendicular  to  the  line;  therefore, 
while  a  line,  which  is  perpendicular  to  a  plane,  has  its 
projections  perpendicular  respectively  to  the  traces  of  the 
plane,  two  lines  which  are  perpendicular  to  each  other 
do  not  necessarily  have  their  projections  perpendicular. 
For  by  reference  to  Figure  34,  any  line  in  the  plane  T  and 


PROBLEMS   IN   POINT,    LINE   AND   PLANE  57 

passing  through  the  piercing  point  of  the  line  OR  with  the 
plane  T  would  be  perpendicular  to  the  line  OR.  Or  again, 
any  line  in  the  plane  T  would  be  parallel  to  some  line 
which  could  be  drawn  through  the  piercing  point  of  the 
line  OR  with  the  plane  T  and  this  latter  line  would  be 
perpendicular  to  OR  because  it  lies  in  a  plane  perpendic- 
ular to  OR.  Much  confusion  is  apt  to  arise  upon  these 
two  points. 

The  angle  a  line  makes  with  a  plane  is  the  angle  which 
it  makes  with  its  projection  on  that  plane. 

The  angle  between  two  planes  may  be  defined  as  the 
plane  angle  between  two  lines  lying  one  in  each  plane,  and 
perpendicular  to  the  line  of  intersection  of  the  planes  at  a 
common  point.  These  lines  are  also  known  as  the  lines  of 
greatest  declivity  of  the  one  plane  with  respect  to  the 
other. 

62.  THEOREM  VIIL 

If  two  lines  are  perpendicular  to  each  other  in  space 
and  one  of  them  is  parallel  to  a  coordinate  plane  of  pro- 
jection their  projections  on  that  plane  are  perpendicular  to 
each  other. 

Proof: — For  since  one  of  the  lines  is  parallel  to  a 
coordinate  plane,  the  projecting  plane  of  any  other  line 
perpendicular  to  it  will  have  its  trace  with  the  coordinate 
plane  perpendicular  to  the  projection  of  the  first  line, 
but  this  trace  is  the  projection  upon  the  coordinate  plane 
of  the  second  line,  hence  the  projections  of  the  lines 
upon  the  one  coordinate  plane  are  perpendicular  to  each 
other. 


58 


DESCRIPTIVE    GEOMETRY 


63.  To  find  the  angle  between  two  lines  which  intersect. 
Analysis:— T/^e  value  of  the  angle  appears  when  the 
plane  of  the  intersecting  lines  is  either  parallel  to  or  is 
coincident  with  a  coordinate  plane;  hence  if  the  lines  are 
oblique  to  both  coordinate  planes,  revolve  the  vertex  of  their 
angle  about  a  trace  of  the  plane  of  the  lines  until  it  lies  in 
the  coordinate  plane  of  that  trace.  The  traces  of  the  given 
lines  are  fixed  points  in  the  revolution,  hence  are  points  in 
the  sides  of  the  revolved  position  of  the  angle,  whence  the 
true  angle  can  be  obtained. 


W 

Figure   85. 

Construction: — Let  AB  and  CD,  Figure  35,  be  two  lines 
intersecting  in  the  point  0.  Find  the  plane  T  of  the  two 
lines.  E  and  F  are  the  respective  H  traces  of  the  lines 
lying  on  the  H  trace  of  their  plane.  Revolve  the  vertex  O 
about  the  H  trace  of  this  plane  into  H.  The  H  traces  of 
the  lines  are  fixed  points  in  the  revolution  since  they  lie  on 
the  axis,  o^  is  the  revolved  position  of  the  vertex  and 
eoJoT  its  supplement  will  be  the  required  angle. 


PROBLEMS    IN    POINT,   LINE    AND    PLANE  59 

If  one  of  the  lines  is  paraUel  to  a  coordinate  plane,  the 
most  convenient  construction  is  to  revolve  the  other  line 
about  it  until  it  is  parallel  to  the  same  plane,  whence  the 
angle  appears  its  true  value  as  projected  on  that  plane. 

64.  To  draw  the  projections  of  any  desired  division  of 
an  angle  between  two  lines. 

Analysis:— i^mc?  the  true  value  of  the  angle  between  the 
two  given  lines  by  Section  63.  Then  draw  the  revolved 
position  of  any  radial  line  dividing  the  angle  into  desired 
proportional  parts.  The  intersection  of  this  line  with  the 
axis  of  revolution  or  trace  of  the  plane  of  the  lines  is  a  fixed 
point.  Connect  this  with  the  projection  of  the  vertex  of  the 
angle  whence  the  angle  is  divided  in  projection, 

65.  Through  a  point  in  a  given  oblique  plane  to  draw 
two  lines  making  a  given  angle  with  each  other. 

Analysis:— Bevolve  the  given  point  as  the  vertex  of  the 
angle  about  either  trace  of  the  plane  until  it  lies  in  the 
coordinate  plane  of  that  trace.  Draw  two  lines  through  this 
revolved  position  of  the  vertex,  to  intersect  the  trace  of  the 
plane  in  two  fixed  points  in  the  sides  of  the  irrojection  of  the 
required  angle.  Connecting  these  points  ivith  the  projection 
of  the  vertex  gives  one  projection  of  the  angle. 

66.  To  project  a  line  on  any  oblique  plane. 

Analysis: — The  projection  of  a  line  on  a  plane  is  the 
locus  of  the  projections  of  all  points  of  the  line  upon  that 
plane.  Hence  take  any  two  points  of  the  line  and 
draw  perpendiculars  to  the  plane  f  Theorem  VII)  and  find 
their  piercing  points  (by  Section  57).  The  line  joining 
these  piercing  points  ivillbe  the  projection  desired. 


60 


DESCRIPTIVE    GEOMETRY 


If  the  given  line  pierces  the  given  plane  within  the 
limits  of  the  drawing  it  is  only  necessary  to  project  one 
other  point  of  the  line  on  the  plane  since  the  piercing" 
point  is  one  of  the  desired  points  in  the  projection. 

\ 

\l 


rW       .r/\ 


/ 


Figure  36 


Construction:— Let  AB,  Figure  36,  be  the  line  and  T 
the  plane.  From  A  and  B  drop  perpendiculars  AAi  and 
BBi  ,  respectively,  to  the  plane  T.  Find  the  piercing  points 
of  these  perpendiculars,  by  using  their  H  projecting  plane 
in  each  case.  Ai  and  Bi  will  be  the  respective  piercing 
points  whence  a'\'h\  is  the  vertical  projection,  and  a^  b^ 
the  horizontal  projection,  of  the  given  line  on  T. 

If  the  given  plane  is  perpendicular  to  either  coordinate 
plane,  then  the  trace  on  that  plane  to  which  it  is  perpen- 
dicular will  contain  at  once  the  piercing  points  of  the 


PROBLEMS    IN    POINT    LINE    AND   PLANE  61 

perpendiculars  from  the  points  in  the  given  line  to  the 
plane. 

If  the  given  plane  is  parallel  to  the  G,L,^  it  is  neces- 
sary to  use  an  end  plane,  for  the  projecting  perpendiculars 
of  the  line  lie  in  planes  perpendicular  to  the  G.L.  The 
intersections  of  perpendiculars  from  the  points  on  the  end 
projection  of  the  line  to  the  end  trace  of  the  given  plane, 
are  the  projections  of  the  respective  piercing  points  with 
the  plane. 

If  the  given  plane  has  its  traces  in  the  G.L.,  an  end 
plane  is  also  necessary  and  for  similar  reasons. 

67.    To  find  the  angle  a  line  makes  with  a  given  plane. 

Analysis: — Since  hy  definition  the  desired  angle  is  that 
between  the  line  and  its  projection  on  the  plane,  project  the 
given  line  upon  the  given  plane,  then  find  the  value  of  the 
angle  hy  revolving  the  line  and  its  projection  about  either 
trace  of  their  plane  tvith  the  coordinate  plane  into  the  latter, 
whence  the  true  value  appears. 

Construction:— Let  the  line  AB  and  the  plane  T  be 
given  as  in  Figure  37.  Find  the  projection  AiBi  of  AB 
on  the  plane  T  by  Sec.  66,  Find  the  Y  trace  of  the  plane 
of  the  lines  AB  and  A  ^  B  i .  The  Y  trace  of  AB  is  C  and 
that  of  AiBi  is  D.  Revolve  A  and  Ai  about  the  Y  trace 
of  the  plane  of  AB  and  AiBi  into  Y.  A  falls  at  e'  and 
Ai  falls  at  e\.  Hence  the  angle  required  is  that  between 
e'c'  and  e\  d' , 

If  the  given  plane  is  parallel  to  the  G.L.  or  has  its 
traces  in  the  G.L.,  it  is  necessary  to  use  an  end  plane  to 
get  the  projection  of  the  line  upon  the  given  plane. 


62 


DESCRIPTIVE     GEOMETRY 


V  trace  of 


FIGUKE    87. 


68.    To  find  the  angle  between  two  planes. 

Method  No.  1.  Analysis:—^?/  definition,  if  a  plane  is 
passed  perpendicular  to  their  line  of  intersection,  it  will  cut 
a  line  from  each  which  is  a  measure  of  the  angle  required. 
Therefore  assume  a  point  upon  the  line  of  intersection  and 
find  the  traces  of  a  plane  perpendicular  to  the  latter  and 
containing  the  point.  The  points  in  which  either  trace  of 
this  auxiliary  plane  cuts  the  corresponding  traces  of  the 
given  planes  will  he  points  in  the  sides  of  the  required  angle. 
The  assumed  point  on  the  line  of  intersection  of  the  planes  is 
the  vertex  of  the  angle.  By  revolution  into  a  coordinate 
plane,  the  true  value  can  he  ohtained. 

Construction: — Let  two  planes  T  and  S  be  given  as  in 
Figure  38.    Assume  the  point  O  upon  their  line  of  inter- 


PROBLEMS    IN    POINT,  LINE    AND    PLANE  63 


FIGUBE    88. 

section  through  which  to  pass  a  perpendicular  plane.  By 
use  of  a  vertical  of  the  plane  its  H  trace,  PH  is  found,  and 
from  the  intersections  of  PH  with  the  H  traces  of  S  and  T, 
the  points  A  and  B  establish  the  sides  of  the  angle  AOB. 
The  vertex  0  revolved  into  H  about  PH  falls  at  Oi  whence 
aOih  is  the  true  value  of  the  angle  required. 

To  assume  the  point  upon  the  line  of  intersection 
through  which  to  pass  a  perpendicular  plane  does  not 
always  bring  a  trace  of  the  plane  upon  a  convenient  part  of 
the  drawing  to  construct  the  angle,  hence  while  this 
method  is  simple  to  understand  it  is  not  in  general  the 
best. 

Method  No.  2.  Analysis:— ^>^  auxiliary  plane  that  is 
perpendicular  to  the  line  of  intersection  of  the  two  given 
planes  and  cuts  from  each  plane  a  line,  cuts  also  a  line  from 
the  projecting  plane  of  the  line  of  intersection  which  in  turn 
pierces  the  coordinate  plane  in  the  trace  of  the  auxiliary 
plane  with  the  latter.  Hence  a  trace  of  the  auxiliary 
plane  may  he  assumed  and  the  piercing  point  tvith  it  found 
of  the  line  of  intersection  of  the  two  given  planes.    The 


64  DESCRIPTIVE    GEOMETRY 

remainder  of  the  solution  is  the  same  as  for  Method  No.  1. 


Construction:— Let  the  conditions  be  given  again  as 
they  were  in  Figure  38,  see  Figure  39.  Assume  the  H  trace 
of  an  auxiliary  plane  P  perpendicular  to  the  line  of  inter- 
section of  the  planes.*  The  point  q  in  which  this  trace 
cuts  the  H  projection  of  the  line  of  intersection  is  the  H 
projection  of  the  H  trace  of  the  line  OQ  cut  from  the  pro- 
jecting plane  of  the  line  of  intersection  by  the  auxiliary 
plane.  Eevolve  the  line  of  intersection  CD  about  its  H 
projection  into  H  and  at  q  draw  a  perpendicular  to  it  qo^. 
O  is  the  piercing  point  of  the  line  CD  with  the  auxiliary 
plane  P  and  o  is  its  H  projection.  Whence  aob  is  the  pro- 
jection of  the  angle.  By  revolving  the  vertex  0  into  H  at 
O2 » it  is  found  that  ao  2  &  is  the  true  value  of  the  angle. 

If  both  planes  are  perpendicular  to  a  coordinate  plane, 
the  angle  between  their  traces  on  that  plane  is  the  angle 
between  the  planes. 

If  one  of  the  planes  has  its  traces  in  the  G,L.,  or  if 

*Its  corresponding  trace  Is  not  needed  In  the  construction. 


PROBLEMS   IN    POINT,    LINE    AND    PLANE  65 

both  planes  have  their  traces  in  or  parallel  to  the  G.L., 
then  it  is  necessary  to  find  their  intersection  with  an  end 
plane.  In  the  first  case  the  intersection  of  the  traces  of 
both  planes  on  the  end  plane  is  a  point  in  the  line  of  inter- 
section of  the  two  planes.  In  the  latter  two  cases  the 
angle  between  the  traces  on  the  end  plane  is  the  true  angle 
between  the  planes. 

If  the  traces  of  both  planes  intersect  each  other  on  the 
G.L.,  it  is  necessary  to  either  use  auxiliary  planes  perpen- 
dicular to  a  coordinate  plane  or  to  pass  an  auxiliary  plane 
parallel  to  one  of  the  given  planes  in  order  to  get  the  pro- 
jections of  the  line  of  intersection  of  the  planes.  In  the 
latter  case  the  projections  first  obtained  are  those  of  a  line 
parallel  to  the  line  of  intersection. 

If  the  angle  between  any  oblique  plane  and  a  coordinate 
plane  is  desired,  the  auxiliary  plane  will  be  perpendic- 
ular to  the  coordinate  plane  and  hence  cut  from  the 
oblique  plane  a  line  of  greatest  declivity  with  respect  to 
the  coordinate  plane  and  from  the  latter  a  line  of  greatest 
declivity  with  respect  to  the  oblique  plane,  the  angle 
between  which  is^the  desired  angle. 

69.  Given  a  line  in  a  plane,  to  draw  another  plane  inter- 
secting the  first  in  the  given  line  and  making  a  given  angle 
with  the  given  plane. 

Analysis:— T^e  given  line  will  be  the  line  of  intersection 
of  the  two  planes,  hence  to  establish  the  plane  angle  which 
is  given,  through  any  point  of  the  given  line  pass  a  plane 
perpendicular  to  it  and  note  its  intersection  with  the  given 
plane.  Revolve  the  line  of  intersection  about  a  trace  of  the 
auxiliary  perpendicular  plane  into  a  coordinate  plane  and 
construct  the  angle  required.     Revolve  it  bach  again  and 


66 


DESCRIPTIVE    GEOMETRY 


the  second  side  of  the  angle  as  derived  and  the  given  line 
determine  the  vlane  desired. 

70.    To  find  the  angle  between  two  planes  by  the  method 
of  change  of  coordinate  planes. 

Analysis:— 7/  one  coordinate  plane  is  taken  parallel  to 
the  line  of  intersection  of  the  given  planes  and  the  corres- 
ponding one  perpendicular  to  it,  this  line  will  he  projected 
upon  the  latter  coordinate  plane  as  a  point  and  the  angle 
between  the  planes  will  he  projected  upon  the  same  planCy 
its  true  value,  heing  the  angle  between  the  traces. 


Figure   40, 


Construction: — Let  the  planes  T  and  S  be  given  as  in 
Figure  40.  Take  a  Gi  .L  i .  parallel  to  the  H  projection  of 
the  line  CD,  which  is  the  line  of  intersection  of  the  planes, 
giving  c'  id' 1  as  the  V  projection  of  the  line  on  the  Vi 
plane  parallel  to  it;  on  this  plane  also  find  the  V  traces  of 
T  and  S  by  using  C  as  a  point  in  each  plane  and  drawing 


PROBLEMS    IN    POINT    LINE    AND    PLANE 


67 


horizontals  of  the  planes  respectively.  TiV  and  SiV  are 
the  traces  so  found.  Next  take  a  G2.L2.  perpendicular  to 
the  Yi  projection  of  CD  getting  the  new  Hi  projection  c^d  2 
as  the  projection  of  the  line  on  this  plane.  This  is  also  a 
point  in  the  new  H  traces  of  the  planes  T  and  S.  Hence 
the  angle  between  TgH  and  S2H  is  the  true  angle 
between  the  planes. 


Figure   41 


71.    To  find  the  common  perpendicular  of  two  non-in- 
tersecting lines. 

Analysis: — If  an  auxiliary  plane  is  passed  through  one 
of  the  lines  and  parallel  to  the  other  the  common  perpendic- 
ular will  he  projected  on  this  plane  as  a  point ,  i.e.,  the 
point  of  intersection  of  the  one  line  and  the  projection  of  the 


68  DESCRIPTIVE     GEOMETRY 

other.  Hence  at  this  point  erect  a  perpendicular  to  the 
auxiliary  plane  to  meet  the  other  line.  It  will  he  the  line 
desired. 

Construction:  Let  the  lines  AB  and  CD  be  given  as 
shown  in  Fig.  41.  Pass  a  plane  through  the  line  CD  and 
parallel  to  AB,  (by  Sec.  52)  using  an  auxiliary  line  EF 
parallel  to  AB  and  intersecting  CD.  TH  and  TV  are  its 
traces.  Project  AB  on  the  planq.  T  by  using  perpendicu- 
lars from  A  and  B  to  the  plane  T  (by  Sec.  QQ)  their  piercing 
points  are  Ai  and  Bi  respectively.  Aj  Bi  is  the  projection 
of  AB  on  the  plane  T.  At  O,  the  point  of  intersection  of 
the  projections,  draw  the  projections  of  a  perpendicular  to 
the  plane  T.  It  intersects  the  line  AB  in  the  point  P.  OP 
is  the  required  perpendicular  o'px\  is  true  length  of  this 
perpendicular. 

Method  No.  2,  Analysis:— ^^  change  of  coordinate 
planes^  project  the  lines  upon  a  new  V^  plane  tvhich  is  parallel 
to  one  of  theyn  and  upon  a  new  H^  plane  perpendicular  to  the 
same  line.  The  required  distance  is  the  perpendicular 
distance  between  the  two  lines  as  projected  on  H^,  one  of 
them  being  projected  as  a  point. 

72.     Given  one  trace  of  a  plane  and  the  angle  the  plane 
makes  with  that  coordinate  plane,  to  find  the  other  trace. 

Analysis: — If  the  trace  of  an  auxiliary  plane -is  draivn 
which  cuts  out  the  lines  of  greatest  declivity  of  the  coordi- 
nate plane  and  the  given  plane  tvith  respect  to  each  other, 
its  trace  will  he  perpendicular  to  the  given  trace  and  the 
projections  of  the  lines  cut  out  of  each  plane  will  coincide 
ivith  it.  Hence  assume  a  line  of  greatest  declivity  of  the 
given  plane  to  he  revolved  ahout  its  projection  on  the  coordi- 


PROBLEMS    IN    POINT,  LINE    AND    PLANE 


69 


nate  plane,  until  it  lies  in  the  latter.  It  will  then  make 
the  angle  with  its  projection  that  the  required  plane  is  to 
make  with  the  coordinate  plane.  Construct  this  angle  and 
find  the  revolved  position  of  the  piercing  point  with  the  other 
coordinate  plane,  of  the  line  of  greatest  declivity.  The  true 
projected  position  of  this  piercing  point  is  a  point  in  the 
desired  trace. 


\ 


\        X 


\ 


TV 


!P      \o' 


v/L. 


\ 


^- 


./ 


"\ 


y^ 


X^ 


/' 


y 


y 


F  IGUKE    42. 


Construction:— Let  the  trace  TH  of  a  plane  be  given  as 
in  Fig.  42  and  let  its  required  angle  with  H  be  60** ,  SH  is 
the  trace  of  any  auxiliary  plane  perpendicular  to  both  T 
and  H.  The  portion  of  it,  op,  coincides  with  the  H  projec- 
tion of  the  line  cut  from  the  plane  T  and  also  that  of  the 
line  cut  from  H.  Assuming  the  former  to  be  revolved  into 
H,  it  falls  at  Oi9 1 ,  making  an  angle  of  60**  with  op.  The 
point  p  1 ,  being  on  a  perpendicular  from  op  to  the  line  op  i , 
is  the  revolved  position  of  the  piercing  point  of  the  line  of 


70  DESCRIPTIVE    GEOMETRY 

greatest  declivity  of  the  plane  T  with  the  V  plane,  hence 
the  projection  of  the  piercing  point  is  a  distance  above 
the  G.L.  on  a  perpendicular  at  p  that  the  point  p,  is  from 
p.    TV  is  the  required  trace  of  the  plane. 

73.  To  find  the  distance  from  a  point  to  a  line. 

Method  No.  1,  Analysis:— 7/  a  plane  is  passed  tJirougJi 
the  point  perpendicular  to  the  line,  the  distance  from  the 
given  point  to  the  piercing  point  of  the  line,  with  this  plane, 
will  he  the  required  distance. 

Method  No.  2,  Analysis:— Pass  a  plane  through  the 
point  and  the  line,  revolve  both  about  either  trace  of  their 
plane  until  they  lie  in  the  coordinate  plane;  the  perpendicular 
distance  between  point  and  line  will  be  projected  its  true 
length, 

74.  Given  one  trace  of  a  plane  and  the  angle  it  makes 
with  the  corresponding  plane  of  projection,  to  find  the 
corresponding  trace. 

Analysis:— The  plane  will  be  tangent  to  a  right  circular 
cone  whose  apex  is  on  the  given  trace  with  its  base  in  the 
corresponding  coordinate  plane  and  axis  perpendicular  to 
the  latter;  the  elements  of  this  cone  will  make  the  same 
angle  with  the  corresponding  coordinate  plane  that  the 
required  plane  makes.  Hence  assume  any  point  upon  the 
given  trace  and  draw  the  projection  of  the  cone  upon  that 
coordinate  plane;  it  will  be  an  isosceles  triangle  with  base 
in  the  G.L.,  and  base  angles  equal  to  the  given  angle. 
Draw  the  corresponding  projection  of  the  cone  which  will 
be  a  circle.  The  trace  of  the  required  plane  upon  this  cor- 
responding plane  will  be  tangent  to  this  circle. 


PROBLEMS   IN   POINT,    LINE   AND   PLANE 


71 


Construction:— Let  TH  be  given  as  in  Figure  43  and  let 
the  required  angle  with  V  be  60°.  Assume  any  point  on  it 
as  O.  Draw  OP  and  OQ  the  contour  elements  of  the  pro- 
jection of  the  cone  on  H  whose  base  angles  are  60°.  qp  is 
the  H  projection  of  the  base  in  V.  Draw  a  circle  with  o' 
as  a  center  and  o'p'  as  a  radius.  Tangent  to  this  circle 
draw  TV,  the  required  trace. 

Note:— The  problem  could  be  solved  by  considering 
only  the  triangle  formed  by  a  plane  passed  perpendicular 
to  V  and  the  plane  T.  o'o\  is  the  revolved  position  of  the 
line  cut  from  H  and  could  be  drawn  from  o '  in  any  direc- 
tion, o'p/  is  also  the  revolved  position  of  the  line  cut  from 
V  and  will  be  perpendicular  to  o' o\.  Lastly,  the  line 
o' ip' I  is  the  revolved  position  of  the  line  cut  from  the 
plane  T  and  can  be  drawn  to  make  the  required  angle  with 
o'p'  I,  The  required  V  trace,  then,  is  tangent  to  a  circle  of 
radius  o'p/  The  true  revolved  position  of  the  triangle  is 
when  op'  1  is  perpendicular  to  TV. 


TO  DESCRIPTIVE    GEOMETRY 

75.    Given  the  angles  a  plane  makes  with  both  coordinate 
planes  to  locate  its  traces. 

Analysis:— T/^e  required  plane  can  he  considered  as 
tangent  to  a  sphere  of  any  radius  whose  center  is  in  the 
G.L,  If  planes  are  assumed  which  are  perpendicular  to  the 
given  plane  and  to  the  coordinate  planes  respectively,  thus 
cutting  out  lines  from  each  which  will  he  measures  of  the 
angles  the  required  plane  makes  with  the  respective  coordi- 
nate planes,  and  if  further  these  planes  are  passed  through 
the  center  of  the  sphere  they  will  cut  lines  respectively  from 
the  required  plane  which  are  tangent  to  the  sphere.  There- 
fore the  triangles  formed  hy  the  assumed  planes  cutting  the 
required  plane  and  hoth  coordinate  planes  in  each  case  can 
he  assumed  separately — dealt  with  as  the  one  triangle  was 
dealt  with  in  Section  72,  Ohserving  the  fact  that  the 
revolved  position  of  the  lines  cut  from  the  required  plane  in 
each  case  will  he  tangent  to  a  circle  ivhich  is  the  projection 
of  the  sphere  and  have  each  one  vertex  at  the  center  of  that 
sphere. 

Construction.— Let  the  required  angles  of  a  plane  with 
V  and  H  be  a  and  P  respectively.  See  Fig.  44.  Draw  the 
projections  of  a  sphere  with  any  radius  and  center  at  any 
point  O.  Next  draw  an  assumed  revolved  position  of  the 
lines  which  would  be.  cut  from  the  V  plane,  the  required 
plane  and  the  H  plane,  by  a  plane  assumed  to  be  perpen- 
dicular to  the  first  two.  o '  a '  &i  is  the  triangle  made  by  these 
lines,  oa'  is  the  line  cut  from  Y,  a'h^thQ  line  cut  from 
the  required  plane  and  o'h^  the  line  cut  from  the  H  plane. 
h^a' o'  is  made  equal  to  a  and  a'h^i's,  drawn  tangent  to  the 
sphere.  The  V  trace  of  the  plane  is  tangent  to  a  circle 
with  radius  o' a'  and  the  H  trace  goes  through  a  point  h  on 


PROBLEMS    IN    POINT,    LINE    AND    PLANE  73 


Y 


"k 


/' 


FiaUBB    44. 

a  perpendicular  to  the  G.L.  at  O,  and  distant  from  0  equal 
to  o&,  .  Likewise,  and  constructed  similarly,  ocd\  is  the 
triangle  formed  by  lines  cut  from  the  H  plane,  the  required 
plane  and  the  V  plane  by  a  plane  perpendicular  to  the  first 
two,  with  the  angle  5  as  shown.  Ocis  the  line  cut  from 
H,  cd\  ,  the  line  cut  from  the  required  plane  and  06?' i,  the 
line  cut  from  V.  The  H  trace  of  the  required  plane  is  tan- 
gent to  a  circle  of  radius  oc  and  the.  V  trace  goes  through 
the  point  6?',  a  distance  od\  above  H. 

76.    Given  the  angle  a  line  makes  with  both  coordinate 
planes  to  draw  its  projections. 

Note:— In  this  problem  the  analysis  for  the  general  so- 
lution is  difficult  to  follow  hence  a  special  solution  is  added 
after  the  general  form. 

General  Analysis: — Consider  the  line  as  limited  by  its 


74  DESCEIPTIVE    GEOMETKY 

traces.  Assume  either  projecting  plane  of  the  line  to  he 
revolved  into  either  coordinate  plane  about  its  intersection 
with  that  coordinate  plane,  Bevolve  the  other  projecting 
plane  of  the  line  bodily  with  it  into  the  same  coordinate 
plane  and  about  the  line  of  intersection  of  the  two  project- 
ing planes,  which  is  the  line  itself.  The  revolved  position  of 
the  line  is  the  hypotenuse  of  two  right  angled  triangles, 
an  angle  of  each  of  which  being  oppositely  directed,  is  the 
respective  angle  the  line  makes  with  the  coordinate  planes. 

Specific  Analysis:— Assume  a  line  limited  by  its  V  and 
H  traces.  Bevolve  the  H  projecting  plane,  bounded  by  the 
line,  its  H projection  and  the  V  trace  of  the  projecting  plane, 
into  V;  next  revolve  the  V  projecting  plane  of  the  line 
bounded  by  the  line,  its  V  projection  and  the  H  trace  of  the 
projecting  plane,  into  V  about  the  revolved  position  of  the 
line  when  it  lies  in  V.  In  the  first  triangle,  the  angle 
between  the  G.L.  and  the  revolved  position  of  the  line  is  the 
angle  with  H.  Opposite  to  that,  with  the  revolved  position 
of  the  line  as  a  side,  is  the  angle  the  line  makes  with  V, 
from  which,  by  completing  the  triangle,  the  distance  of  its 
H  trace  from  the  G.L.  can  be  found.  Next  revolve  back 
into  proper  projective  position  the  first  triangle,  to  do  which 
the  H  trace  must  be  located  at  the  distance  from  the  G.L. 
which  it  is  found  to  be  from  the  revolved  position  of  the 
second  triangle. 

Construction:  Assume  oa'b^  ,  Fig.  45,  to  De  the  re- 
volved position  of  the  H  projecting  plane  of  a  line  of 
which  a '&i  is  the  line  itself.  On  the  latter  as  an  axis, 
assume  the  V  projecting  plane  to  be  revolved  into  Y.  The 
perpendicular  distance  of  c'  i  from  a'biis  the  revolved  pos- 
ition of  the  distance  of  the  H  trace  from  the  V  plane,  i.  e. 


PROBLEMS    IN    POINT    LINE    AND    PLANE 


75 


from  the  G-.  L.  Next,  lay  off  the  distance  c'l  &i  on  H  from 
the  G.  L.  and  draw  a  horizontal  line.  Where  an  arc  of  a 
circle,  with  radius  ahi  and  center  at  a  cuts  this  horizontal 
in  h,  is  the  H  projection  of  the  H  trace  of  the  line  and  ah  is 
the  H  projection  of  the  required  line.  The  V  projection  is 
h'a\ 


77.    To  draw  a  regular  pyramid  of  given  base  and  altitude 
with  its  base  in  a  given  oblique  plane. 

Analysis:— r^.y'  Assume  the  base  as  revolved  about 
either  trace  of  the  given  plane  until  it  lies  in  the  coordinate 
plane  of  that  trace,  where  it  can  he  drawn  its  true  shape. 
Hevolve  it  hack  again  into  the  given  plane  hy  any  convenient 
method.  Next,  to  find  the  projections  of  the  apex  of  the 
pyramid,  draw  a  perpendicular  to  the  plane  at  the  center  of 
the  base,  and  hy  any  convenient  method,  get  the  projected 
distance  laid  off  on  this  perpendicular,  i.e.,  the  projection 
of  the  altitude  of  the  pyramid.  The  apex  connected  with 
the  corners  of  the  base  completes  the  pyramid. 

Anaiy sis:— fJ2. J    Assume  the  pyramid  to  he  projected 


76  DESCRIPTIVE     GEOMETRY 

upon  a  plane  perpendicular  to  that  of  the  base,  i,e.,  parallel 
to  the  axis.  Assume  this  projection  as  revolved  about  the 
trace  of  its  plane  with  the  coordinate  plane  until  it  lies  in 
the  latter.  Here  the  axis  will  appear  its  true  length,  since 
it  lies  in  the  coordinate  plane  the  base  will  be  projected  as  a 
straight  line.  By  revolving,  next,  the  base  about  its  inter- 
section with  the  coordinate  plane  into  the  latter,  its  true 
shape  will  appear  and  the  relative  position  of  its  vertices  be 
ascertained.  By  counter  revolution  the  projection  of  the 
pyramid  may  be  constructed. 

Construction:— According  to  Analysis  2.  Let  a  hexa- 
gonal pyramid  with  diameter  of  base  of  2  inches  and 
altitude  2|  inches,  have  its  base  in  any  oblique  plane  T, 
see  Figure  46,  and  an  edge  of  the  base  lying  in  H. 
Assume  the  pyramid  to  be  projected  on  any  plane  S  per- 
pendicular to  H  and  to  T.  The  line  ab^,  is  the  revolved 
projection  of  the  base  on  S,  being  also  the  revolved  posi- 
tion of  the  line  cut  from  T  by  S  and  p^  is  the  apex  also 
revolved  into  H.  Next  assume  the  base  of  the  pjn^amid  to 
be  revolved  about  a&  i ,  as  a  diameter,  giving  1 1 ,  2 1 ,  3 1 , 
etc.,  as  the  vertices  of  the  base  In,  2n,  3„,  etc.,  are  the 
true  projected  positions  of  the  vertices  on  the  plane  S  when 
the  latter  is  revolved  into  H.  The  horizontal  distances  of 
the  vertices  from  the  diameter  are  constant,  hence  the 
edge  of  base  in  H  can  be  assumed  anywhere  and  by 
counter  revolution  the  vertices  1  ,2  ,3,  etc.,  found.  The 
vertical  projection  of  the  pyramid  is  obtained  by  noting 
that  the  distances  of  points  above  the  G.L.  are  respectively 
equal  to  the  distances  of  the  vertices  1 1 ,  2i ,  3 1 ,  etc.,  above 
SH. 

It  is  to  be  noted  that  the  method  just  described  is 


PEOBLEMS   IN    POINT,    LINE    AND    PLANE 


77 


FiGUBE    46. 

really  that  of  a  change  of  coordinate  planes  where  SH  is 
the  G.L.  of  a  new  Vi  plane  parallel  to  the  axis  of  the 
pyramid. 

78,    To  draw  a  circle  through  any  three  given  points. 

Analysis:— T/^e  three  points  are  the  extremities  of  chords 
of  the  circle  passing  through  them.    Hence  find  the  2)lane  of 


78 


DESCRIPTIVE    GEOMETRY 


tJie  three  points;  revolve  them  ahout  either  trace  of  their 
plane  into  the  coordinate  plane  and  draw  the  circle  passing 
through  them.  Bevolve  the  circle  into  its  true  projective 
position  hy  any  convenient  method. 

Note:— A  convenient  method  is  to  use  parallel  chords 
perpendicular  to  the  trace  of  the  plane  about  which  the 
three  points  are  revolved. 


Figure    47. 


PROBLEMS    IN    POINT    LINE    AND   PLANE  79 

Construction:— Let  the  three  points  A,  B  and  C  be 
given  as  in  Figure  47.  The  traces  of  their  plane  are  TH 
and  TV  respectively.  Revolve  the  points  about  the  H 
trace  of  their  plane  into  H  to  fall  at  a i ,  &i  and  Ci ,  respect- 
ively, whereupon  the  circle  passing  through  them  may 
be  constructed. 

To  revolve  the  circle  back  into  its  true  projective 
position,  the  following  method  may  be  employed.  Pass  a 
series  of  auxiliary  planes  Q,  R,  S,  etc.,  perpendicular  to  the 
H  plane  and  the  plane  T  to  cut  the  circle  lying  in  the  latter 
plane,  each  in  two  points.  The  lines  of  intersection  of 
these  several  planes  with  T,  will  pass  through  these  points, 
respectively ;  further,  the  points  revolved  into  H  about  TH, 
will  lie  on  a  perpendicular  to  TH  as  shown.  By  counter 
revolution  these  may  be  located  on  the  respective  lines  of 
intersection  of  the  auxiliary  planes  with  T. 

79.  Through   a  given   point,  to   draw  a    line  making  a 
given  angle  with  a  given  oblique  plane. 

Analysis: — The  locus  of  the  lines  which  can  he  drawn 
from  the  given  point  to  the  given  plane j  making  a  certain 
angle  with  it,  form  the  elements  of  a  right  circular  cone 
whose  hase  is  in  the  plane  and  whose  apex  is  the  point. 
Therefore  draw  the  cone  with  elements  at  the  required  angle 
with  the  base  hy  a  similar  method  to  that  for  the  hexagonal 
pyramid  in  Section  77,  Choosing  any  one  element,  revolve 
it  hack  into  true  projective  position.  The  construction  is 
left  to  the  student. 

80.  Through  a  given  line,  to  pass  a  plane  perpendicular 
to  a  given  plane. 

Analysis:— The  required  plane  must  contain  a  perpcn 


80  DESCRIPTIVE     GEOMETRY 

dicular  to  the  given  plane,  hence,  through  any  point  of  the 
given  line,  draw  a  perpendicular  to  the  given  plane.  The 
plane  of  the  two  lines  is  the  plane  required.  The  construc- 
tion is  left  to  the  student. 

81.    LociL 

When  a  point  moves  so  as  to  satisfy  a  given  condition 
or  conditions,  the  path  it  traces  is  called  its  locus  under 
these  conditions. 

As  an  example,  the  parabola  is  the  locus  of  a  point 
moving  in  a  plane  according  to  the  conditions  of  the  equa- 
tion y''=4px.  This  may  be  defined,  in  graphical  terms, 
as  the  locus  of  a  point  which  moves  in  a  plane  so  that  its 
distance  from  a  fixed  line  is  equal  to  its  distance  from  a 
fixed  point.  The  line  is  known  as  the  directrix,  and  the 
point  as  the  focus  of  the  parabola.  The  ratio  of  the  two 
distances  is,  in  mathematics,  called  the  eccentricity  (e) ;  for 
the  parabola,  it  is  equal  to  unity.  It  has  different  values 
for  the  different  conies. 

The  ellipse  is  the  locus  of  a  point,  which  moves  in  a 
plane,  so  that  the  ratio  of  its  distance  from  a  fixed  line  to 
its  distance  from  a  fixed  point  is  greater  than  unity,  in 

X^  11^ 

other  words  satisfies  the  equation  -^  +  7^  =  1. 

a  0 

The  hyperbola  is  the  locus  of  a  point  which  moves  in  a 
plane,  so  that  the  ratio  of  its  distance  from  a  fixed  line  to 
its  distance  from  a  fixed  point  is  less  than  unity,  in  other 

x^        v^ 
words  satisfies  the  equation  -^  —  -?2  =  1. 

a  0 

The  circle,  defined  in  the  same  terms,  is  the  locus  of  a 
point,  which  moves  in  a  plane,  so  that  the  ratio  of  its 
distance  from  a  fixed  line  to  its  distance  from  a  fixed 


PROBLEMS   IN   POINT,   LINE   AND   PLANE  81 

point  is  equal  to  infinity.  In  more  generally  understood 
language,  it  is  the  locus  of  a  point,  which  moves  in  a  plane, 
so  as  to  be  equally  distant  from  a  fixed  point.  Graphical 
locii  are  a  legitimate  part  of  descriptive  geometry. 

82.  To  draw  the  locus  of  points  equally  distant  from  two 
given  planes. 

Analysis:— T/^e  locus  of  points  equally  distant  from  two 
straight  lines  in  a  plane  is  the  bisector  of  the  angle  between 
the  two  lines,  hence  the  locus  of  points  equally  distant  from 
two  planes,  each  containing  one  of  the  lines,  is  the  plane 
bisector  of  the  dihedral  angle  between  the  two  planes. 
Therefore  find  the  angle  between  the  two  planes,  bisect  it 
and  pass  a  plane  through  the  bisector  and  the  line  of  inter- 
section of  the  planes.    This  plane  ivill  be  the  required  locus. 

83.  To  draw  the  locus  of  points    in   space  equi-distant 
from  three  given  points. 

Analysis:  -  The  locus  of  points  in  space,  equally  distant 
from  two  points,  is  a  plane  perpendicular  to  and  bisecting 
the  straight  line  connecting  the  tivo  points.  Hence  the  locus 
of  points,  equally  distant  from  three  points,  will  be  the 
common  line  of  intersection  of  three  such  planes.  Therefore 
find  the  plane  of  the  three  points  and  by  revolving  them 
about  either  trace  of  their  plane  find  the  perpendicular 
bisectors  of  the  lines  connecting  the  points;  these  will  meet 
in  a  point  which  is  the  center  of  the  circle  passing  through 
the  three  points.  Find  the  true  projective  position  of  this 
point  in  the  plane.  Through  it  draw  a  line  perpendicular 
to  the  plane  of  the  points  which  will  be  the  locus  required. 


82  DESCRIPTIVE    GEOMETRY 

REVIEW  QUESTIONS 

If  two  lines  are  parallel  in  space,  are  their  projections 
parallel  ? 

If  two  lines  are  perpendicular  to  each  other  in  space, 
are  their  projections  perpendicular? 

If  two  planes  are  parallel  to  each  other  in  space,  are 
their  respective  traces  parallel  ? 

If  two  planes  are  perpendicular  in  space,  are  their 
respective  traces  perpendicular  to  each  other  ? 

If  a  line  is  parallel  to  a  plane,  are  its  projections 
parallel  respectively  to  the  traces  of  the  plane  ? 

If  a  line  is  perpendicular  to  a  plane,  are  its  projections, 
respectively  perpendicular  to  the  traces  of  the  plane  ? 

What  is  the  locus  of  points  equi-distant  from  a  given 
straight  line  1 

What  is  the  locus  of  lines  which  make  a  constant 
angle  with  a  straight  line  at  a  common  point  in  it  ? 

What  is  the  locus  of  points  in  space  equi-distant  from 
a  fixed  point ! 

What  is  the  locus  of  points  in  space  equi-distant  from 
two  intersecting  lines  I 


CHAPTER   IV. 


GENERATION   AND   CLASSIFICATION   OF   LINES 
AND   SURFACES. 


84.    A  line  is  mathematically  the  path  of  a  moving  point 

under  fixed  conditions,  i.e.,  the  locus  of  the  point  under 
those  conditions ;  it  may  also  be  an  element  of  a  surface 
which  latter  also  has  a  mathematical  interpretation  as  the 
path  traced  by  a  moving  line  under  certain  conditions. 

A  line  may  also  he  defined,  according  to  various 
properties  of  the  line,  one  of  which,  for  example,  is  the 
intersection  of  two  surfaces  or  solids. 

A  straight  line,  or  right  line,  is  the  path  of  a  point 
which  moves  in  a  fixed  direction,  or  again,  the  intersection 
of  two  planes. 

Any  two  consecutive  positions  of  a  moving  point, 
being  infinitely  close,  may  be  regarded  as  the  extremities 
of  an  infinitesimal  element  of  the  path  traced,  or  an 
elementary  line  of  it,  and  in  going  from  the  one  position  to 
the  next,  no  matter  what  the  ultimate  direction  of  the 
path,  the  elementary  path  traced  is  straight. 

All  lines  may  be  grouped  into  three  classes  according 
to  the  law  governing  the  motion  of  the  generating  point. 
(1.)  Eight  lines.  (2.)  Single  curved  lines.  (3.)  Double 
curved  lines. 


84  DESCBIPTIVE     GEOMETRY 

If  a  point  moves  in  a  fixed  direction,  the  path 
generated  is  a  straight  line. 

If  a  point  moves  in  a  constantly  changing  direction 
the  path  is  a  curve;  whether  it  be  a  curve  of  single  curva- 
ture or  a  curve  of  double  curvature  depends  upon  the 
following  conditions : 

If  any  four  consecutive  points,  that  is,  points  in  the 
curve  infinitely  near  to  each  other,  or  any  three  con- 
secutive elements,  that  is,  right  lines  connecting  the 
consecutive  points  as  defined,  lie  in  the  same  plane,  then 
all  the  points  of  the  path  lie  in  the  same  plane  and  the 
curve' is  a  curve  of  single  curvature,  i.e.,  a  plane  curve. 

If  no  four  consecutive  points  nor  any  three  consecutive 
elements,  passing  through  the  four  points,  lie  in  the  same 
plane  then  the  curve  is  a  curve  of  double  curvature,  i.e., 
it  is  not  a  plane  curve. 

A  curve  of  any  kind  may  be  further  considered  as  gen- 
erated by  the  motion  of  a  line  intersecting  itself  in  succes- 
sive positions  in  which  case  the  curve  is  known  as  the  en- 
velope of  the  successive  positions  of  the  line. 

For  example,  if  two  intersecting  lines  as  in  Figure  48, 
are  divided  into  a  number  of  equal  parts  and  the  division 
on  one  farthest  from  the  vertex  is  joined  to  the  first 
division  from  the  vertex  on  the  other  and  so  on  until  all 
the  divisions  are  connected  by  lines,  it  will  be  seen  that 
these  lines  are  tangent  to  a  curve.  This  curve  is  the 
parabola  and  is  the  envelope  of  its  tangents. 

A  curve  of  double  curvature  may  be  considered  as 
generated  by  the  motion  of  a  plane  whose  successive 
positions  intersect  each  other  in  lines  which  constitute  suc- 
cessive elements  of  the  curve.    For  example,  just  previously 


LINES    AND    SURFACES    CLASSIFIED 


85 


Figure  48 . 

it  was  stated  that  if  no  four  consecutive  points,  or  three  con- 
secutive elements  connecting  these  points,  lie  in  the  same 
plane  the  curve  is  a  curve  of  double  curvature.  Conceive 
any  four  such  points  connected  by  elements,  one  through 
points  1  and  2,  another  through  2  and  3  and  a  third 
through  3  and  4.  A  plane  can  be  passed  through  the  first 
and  second  elements,  and  another  through  the  second  and 
third,  but  the  two  planes  will  intersect  each  other  in  the 
second  element,  i.e.,  2  and  3  and  this  is  an  element  of  the 
curve  also ;  in  like  manner  the  successive  planes  will  inter- 
sect each  other  two  and  two  in  successive  elements  of  the 
curve. 

To  sum  up  the  classification  of  lines  and  give  illustra- 
tive examples  of  them,  however  by  no  means  complete, 
the  following  table  is  appended:  — 


85.     Table  of  classification  of  lines. 


86 


DESCRIPTIVE    GEOMETRY 


^ 

eg 

II 

i 

0) 

jg         ?H 

•3.& 

<» 

•  rH 

1 

O         ^ 

^ 

^ 

2    ® 
S-2 

C3 
O 

i 

^    o 

cS 

o 

a; 

J3     o 

<D 

nd 

-^     ./T 

^    ^ 

0) 

2 

llipse 
arabol 

a 
•1— 1 

2 

ogarit 
^yperb 

4^ 

> 
I— 1 

o 

C3     S 

"2 

1.1 

S 

^  PM 

m 

^ 

^  m 

GQ 

H  OQ 

Q 

EH 

. 

" 

.2 

(—1 

.B 

.a" 

1;^ 

§ 

o 

§ 

CO 

13 

a 

o 
o 

CO 

1 

o 

^1 

•a 

•  1— 1 

.1 

eg 

^3 

§ 

^3       rt 

o 

ft 

p-^ 

?-i 

o 

?:; 

as 

Q 

^ 

H 

H 

p^ 

EH 

02 

1 

i-:i 

'S 

' 

'S 

s 

<D 

0) 

1 

1 

CO 

m 

Q 

QQ 

2 

S 

•  rH 

1 

^ 
^ 

-R 

o; 

2 


LINES    AND    SURFACES    CLASSIFIED  87 

86.   Definitions. 

The  conies  have  been  defined  as  the  locii  of  moving 
points.  They  are  also  known  from  their  properties  as 
sections  of  a  cone. 

A  circle  is  the  section  of  a  right  circular  cone  made  by 
a  plane  perpendicular  to  its  axis. 

An  ellipse  is  the  section  of  a  right  circular  cone  made 
by  a  plane  at  a  greater  angle  to  the  axis  than  that  made  by 
the  elements  of  the  cone. 

A  parabola  is  the  section  of  a  right  circular  cone  made 
by  a  plane  at  an  angle  to  the  axis  equal  to  that  made  by 
the  elements  of  the  cone. 

A  hyperbola  is  the  section  of  a  right  circular  cone  made 
by  a  plane  at  an  angle  to  the  axis  less  than  that  made  by 
the  elements  of  the  cone. 

*  The  roulettes  are  a  class  of  curves  traced  by  a  point 
upon  one  curve  when  it  rolls  upon  another  curve.  The 
cycloid,  as  generally  understood,  is  the  path  traced  by  a 
point  on  a  circle,  when  it  rolls  upon  a  straight  line,  hence 
belongs  to  the  roulettes. 

When  both  the  rolling  and  fixed  curves  are  circles,  the 
paths  traced  are  called  trochoids.  The  epicycloid  is  the 
path  traced  by  a  point,  on  a  circle  which  rolls  upon  the  out- 
side of  another  circle.  The  hypocycloid  is  the  path  traced 
by  a  point  on  a  circle  which  rolls  upon  the  inside  of  another 
circle. 

The  rolling  circle  is  called  the  generatrix  and  the  fixed, 
circle  the  directrix. 

The  helix  is  defined  as  a  curve  traced  by  a  point  which 
moves  at  a  uniform  rate  around  a  cylinder  or  cone,  and  at 

*  From  F.  N.  Willson's  classification  in  '  Theoretical  and  Practical  Graphics.' 


88  DESCRIPTIVE    GEOMETRY 

the  same  time  at  a  uniform  rate  in  a  direction  parallel  to 
the  axis  of  the  cylinder  or  cone.  It  is  also  defined  as  the 
shortest  line  that  can  be  drawn  upon  a  cylinder  or  cone, 
between  two  points,  which  lie  neither  upon  the  same  right 
section  or  upon  the  same  right  line  element.  The  term 
cylinder  is  not  limited.  The  shortest  line  which  could  be 
drawn  in  such  a  manner  upon  a  cone  would  be  known  as  a 
conical  helix. 

The  projection  of  a  conical  helix  upon  the  base  of  the 
cone,  when  it  is  a  right  circular  cone,  is  the  archimedian 
spiral  mentioned  at  the  head  of  the  list  of  spirals. 

87.     Every  surface  can  be  generated  by  the  motion  of  a 
line;  the  line  may  be  straight  or  curved. 

All  surfaces  may  be  classified  as  ruled  or    double 
curved.    A  ruled  surface  is  generated  by  the  motion  of  a 
straight  line.    A  double  curved  surface  is  generated  only 
by  the  motion  of  a  curve. 

Ruled  surfaces  are  further  divided  into  three  classes, 
according  to  the  law  of  motion  of  the  generating  line: 
1.  Planes,  2.  Single  curved  surfaces,  3.  Warped  surfaces. 

A  plane  is  unique  and  may  be  defined  as  generated  by 
a  straight  line  moving  so  that  it  touches  another  straight 
line  and  has  its  successive  positions  parallel  one  to  another, 

A  single  curved  surface  is  defined  as  generated  by  a 
straight  line  moving  so  that  any  two  but  no  three  of  its 
consective  positions  are  in  the  same  plane.  These  surfaces 
are  further  divided  into  three  groupes,  according  to  the 
way  in  which  the  elements  intersect  each  other;  the  cyl- 
inders, in  which  a  common  point  of  intersection  of  the 
successive  elements  is  at  infinity;  the  cones,  in  which  a 
common  point  of  intersection  of  the  successive  elements  is 


LINES    AND    SURFACES    CLASSIFIED  89 

in  a  finite  position;  the  convolutes,  in  which  the  elements 
intersect  two  and  two,  no  three  intersecting  in  a  common 
point.  These  latter  are  formed  by  a  line  moving  so  that  it 
is  tangent  to  a  curve  of  double  curvature. 

A  notable  characteristic  of  these  surfaces,  in  distinc- 
tion to  warped  surfaces  and  to  double  curved  surfaces,  is 
that  they  can  be  developed,  that  is  they  can  be  unrolled 
and  laid  out  flat,  for  the  plane  of  any  two  consecutive 
elements  can  be  rolled  upon  one  of  the  elements  until  it 
coincides  with  the  plane  formed  by  this  element  and  the 
next  consecutive  element. 

Warped  surfaces  are  generated  by  a  straight  line  mov- 
ing so  that  no  two  of  its  successive  positions  intersect  each 
other.  The  motion  of  the  line  may  be  controlled  in  several 
ways  and  an  infinite  variety  of  warped  surfaces  obtained. 
This  class  of  surfaces  will  be  discussed  more  fully  a  little 
later.* 

The  warped  surfaces  are  distinguished  from  the  single 
curved  surfaces  and  are  like  the  double  curved  surfaces  in 
that  they  cannot  be  developed,  for  they  do  not  consist  of  a 
succession  of  intersecting  planes. 

Double  curved  surfaces  are  generated  only  by  the  mo- 
tion of  curves  and  have  no  straight  line  elements.  The 
most  familiar  double  curved  surfaces  are  those  which  are 
also  surfaces  of  revolution,  shortly  to  be  defined,  namely, 
the  sphere,  ellipsoid  of  revolution,  paraboloid  of  revolution, 
torus,  etc. 

Surfaces  of  revolution  are  generated  by  the  revolution 
of  one  line,  curved  or  straight,  about  a  straight  line  as  an 


*A  warped  surface  may  also  be  generated  by  the  motion  of  a  curve  which 
changes  its  shape  according  to  a  certain  law. 


90  DESCRIPTIVE     GEOMETRY 

axis.  This  class  does  not  belong  to  any  one  alone  of  the 
groups  before  mentioned,  but  it  partakes  of  their  proper- 
ties. For  example,  if  a  right  line  revolves  about  another  line, 
to  which  it  is  parallel,  it  will  generate  a  cylinder,  which 
is  both  a  single  curved  surface  and  one  of  revolution.  If 
a  right  line  revolves  about  another  line  which  it  does  not 
intersect  it  will  generate  what  is  known  as  a  hyperboloid  of 
revolution,  which  is  both  a  surface  of  revolution  and  a 
warped  surface.  The  latter  is  the  only  surface  which  has 
both  of  these  characteristics.  This  particular  surface  may 
also  be  generated  by  the  revolution  of  an  hyperbola  about 
its  transverse  axis.  If  a  curve  is  revolved  about  a  straight 
line  as  an  axis  it  will,  with  the  exception  above  mentioned, 
generate  a  double  curved  surface  of  revolution  as  the 
sphere,  the  ellipsoid  of  revolution,  the  paraboloid  of  rev- 
olution, hyperboloid  of  revolution,  etc. 

In  fact  surfaces  of  revolution  may  be  classified  under 
ruled  surfaces  and  double  curved.  Under  the  ruled,  come 
only  the  cylinders  and  cones,  and  the  hyperboloid  of  revo- 
lution, the  two  former  being  single  curved  surfaces,  the 
latter  warped.  Under  double  curved,  come  all  those  gen- 
erated by  the  revolution  of  a  curve,  an  infinite  number, 
varying  according  to  the  shape  of  the  curve. 

To  sum  up  the  general  classification,  first,  of  surfaces 
as  a  whole,  then  of  surfaces  of  revolution,  giving  illustra- 
tive examples,  not  complete  however  in  all  cases,  the 
following  tables  are  appended: 


LINES    AND    SURFACES    CLASSIFIED 


91 


88.     Table  of  Classification  of  Surfaces. 

^  Planes 


Surfaces 


Ruled 


Double  curved 
including  sur- 
faces of  revo- 
lution 


{Cylinder 
Cones 
Convolutes 


Warped,   in- 
cluding one 
surface     of 
revolution 


Hyperbolic 

paraboloid 
Right  conoid 
Oblique  conoid 
Helicoid 
Hyperboloid  of 
revolution,  etc. 


Variable  ellipsoid 
Variable  paraboloid,  etc. 


89.    Table  of  Classification  of  Surfaces  of  Revolution. 

Single  f  Right  circular  cylinder 

curved! Right  circular  cone 


Surfaces 

of 
Revolution 


Ruled 


Warped      \  Hyperboloid  of  revolution 


-  Double  curved 


Sphere 
Ellipsoid 
Paraboloid 
Hyperboloid 
.  Torus,  etc. 


90.    The  method  of  generation  of  particular  surfaces  is 

not  limited  to  that  which  has  just  been  described.  For 
illustration  a  cone  may  be  generated  by  a  right  line  pass- 
ing through  a  fixed  point  and  moving  so  as  to  touch  a 


92  DESCRIPTIVE    GEOMETRY 

curve.  Or  again  by  the  motion  of  a  plane  curve  which 
changes  its  shape  according  to  a  certain  law  so  that  all  of 
its  points  describe  rectilinear  paths  meeting  in  a  common 
point. 

Although  the  double  curved  surfaces,  with  which  we 
are  familiar,  are  also  surfaces  of  revolution,  still  among 
the  infinite  forms  of  such  surfaces  which  are  not,  are  several 
notable  ones,  for  example  the  ellipsoid  of  unequal  axes, 
i.e.,  a  surface  formed  by  the  motion  of  a  variable  ellipse 
about  one  of  its  axes  such  that  its  points  describe  elliptical 
paths  whose  centers  are  in  the  axis,  and  also  the  variable 
paraboloid  similarly  formed  by  a  variable  parabola  moving 
about  its  axis,  each  point  describing  an  elliptical  path 
whose  center  is  in  the  axis.  In  these  forms,  sections 
perpendicular  to  the  fixed  line  or  axis  are  ellipses. 

91.  A  curve  is  projected  by  projecting  points  in  the 
curve.  All  curves  whether  of  single  or  double  curvature 
are  projected  as  curves  of  single  curvature  because  all 
their  points  lie  in  the  coordinate  planes  respectively.  A 
curve  may  he  indeterminate  like  a  straight  line,  when  its 
plane  is  perpendicular  to  both  coordinate  planes.  If  a 
curve  is  projected  upon  a  plane,  to  tvhich  it  is  parallel,  it 
will  be  a  parallel,  and  equal  curve.  The  traces  of  a  curve 
are  determined  in  the  same  manner  as  those  of  a  straight 
line. 

92.  THEOREM  IX. 

Two  projections  of  a  curve  being  given,  the  curve  will, 
in  general,  be  completely  determined. 

Proof:— For,  if  at  the  horizontal  projection  of  any  point 


LINES    AND    SURFACES    CLASSIFIED  93 

on  the  curve,  a  perpendicular  is  erected,  to  the  H  plane  it 
will  contain  a  point  of  the  curve  and  similarly  if  at  the 
vertical  projection  of  the  same  point,  a  perpendicular  is 
erected  to  the  V  plane  it  will  contain  the  same  point  of 
the  curve,  hence  the  point,  being  common  to  two  lines, 
must  lie  at  their  intersection,  and  so  for  any  other  point 
of  the  curve. 

93.  One  line  is  tangent  to  another  when  it  has  at  least 
two  consecutive  points  in  common  with  it. 

Assume  a  secant  to  any  curve,  which  cuts  it  in  two 
points,  to  be  moved,  by  rotation  or  translation,  until  these 
two  points  of  intersection  approach  one  another  and 
become  coincident  points ;  the  secant  approaches  tangency 
and  become  such  when  the  two  points  become  coincident. 

A  secant  may,  in  some  cases,  be  moved  until  more 
than  two  points  become  coincident.  This  may  be  illus- 
trated by  a  secant  which  cuts  a  curve,  having  a  point  of 
inflection,  between  the  points  common  to  it  and  the 
secant.  Further,  a  straight  line,  may  be  tangent  to  a 
curve  at  two  or  more  consecutive  points,  and  also  tangent 
elsewhere  at  two  or  more  other  consecutive  points  or  cut 
the  curve  again  in  one  or  more  points. 

In  analystic  geometry,  the  degree  of  the  equation  to 
any  curve,  is  an  index  of  the  maximum  number  of  points 
in  which  a  straight  line  may  cut  the  curve.  Of  course, 
some  points  may  be  imaginary. 

If  a  curve  and  a  straight  line  continually  approach  one 
another^  but  do  not  meet,  they  have  a  point  in  common ; 
i.e.,  are  tangent  at  infinity.  In  this  case,  the  straight 
line  is  said  to  be  an  asymptote  to  the  curve  or  they  are 
asymptotic  to  each  other. 


94  DESCRIPTIVE    GEOMETRY 

Two  curves  are  tangent  to  each  other  when  they  have 
at  least  two  points  in  common  or,  at  a  pair  of  common 
consecutive  points,  they  have  a  common  tangent. 

If  a  right  line  is  tangent  to  a  curve  of  single  curva- 
ture^ it  will  be  contained  within  the  plane  of  the  curve,  for 
by  definition  of  a  curve  of  single  curvature,  if  two  consecu- 
tive points  be  connected  by  a  straight  line,  these  points 
lying  in  the  plane,  the  straight  line  connecting  them  lies 
wholly  in  the  plane. 

If  a  right  line  is  tangent  to  another  right  line  it 
coincides  with  it. 

94.  THEOREM    X. 

If  two  lines  are  tangent  in  space  their  projections  on 
the  same  plane  are  tangent  to  each  other. 

Proof:— For  the  two  or  more  consecutive  points  in 
which  the  lines  are  tangent  to  each  other  are  projected 
as  consecutive  points  of  the  projection  of  both  lines, 
hence  fulfill  the  conditions  of  a  tangent  to  both  lines. 

The  converse  of  this  is  not  true  unless  the  consecutive 
points  in  projection  are  the  projections  of  the  same  points 
of  each  line. 

Although  analytically  a  tangent  touches  a  curve  in 
two  or  more  points,  it  is  customary  to  speak  of  the  *  point 
of  tangency.' 

95.  A  normal  to  a  curve  is  a  line  normal  to  the  tangent 
to  the  curve  at  the  point  of  tangency. 

An  infinite  number  of  perpendiculars  can  be  drawn  to 
a  line  at  a  given  point  constituting  a  perpendicular  plane  at 


LINES    AND    SURFACES    CLASSIFIED  95 

the  point,  hence  a  plane  perpendicular  to  a  tangent  to  a 
curve  at  the  point  of  tangency  will  contain  the  normals 
which  can  be  drawn  to  the  curve  at  that  particular  point. 
But  the  normal  to  a  curve  of  single  curvature  is  generally 
understood  to  mean  that  normal  which  lies  in  the  plane  of 
the  tangent  and  the  curve,  and  will  be  so  spoken  of  in  this 
book. 

If  a  curve  of  single  curvature  is  rolled  upon  its  tangent 
until  a  given  linear  distance  upon  it  comes  successively 
into  contact  with  .the  tangent,  the  curve  is  said  to  be 
rectified. 

96.    To  draw  a  tangent  to  an  irregular  plane  curve  at  a 
point  on  the  curve. 


FIGUKE     49. 


Let  the  curve  be  given  as  in  Fig.  49,  and  the  point  of 
tangency  P.  Draw  a  series  of  chords  to  the  curve  through 
P,  then  with  P  as  a  center  and  any  radius  draw  an  arc  of 
an  auxiliary  circle  to  cut  these  cords.  Lay  off  the  length 
of  each  chord,  (i.e.,  the  distance  from  P  to  the  curve) 
upon  the  chord  produced  and  measuring  from,  and  on  the 
same  side  of,  the  auxiliary  circle  that  the  chord  lies  with  re- 
spect to  the  point  P.  Through  these  points  draw  a  smooth 
curve.  Where  it  intersects  the  auxiliary  circle  will  be  a 
point  in  the  tangent. 


96 


DESCRIPTIVE    GEOMETRY 


97.    To  find  the  approximate  point   of   tangency   of   an 
irregular  plane  curve  and  its  tangent 


Figure  50. 


Let  the  curve  and  its  tangent  be  given  as  in  Fig.  50. 
Draw  a  series  of  chords  of  the  curve  parallel  to  the  given 
tangent.  At  the  intersection  of  these  chords,  respectively, 
with  the  curve,  erect  perpendiculars  to  the  chords  and  lay- 
off upon  them,  respectively,  distances  measured  in  opposite 
directions  equal  in  length  to  the  chords  terminating  in  the 
perpendiculars.  Draw  a  smooth  curve  through  the  points 
located  and  it  will  intersect  the  given  curve  in  the  point  of 
tangency. 

The  two  preceding  constructions  are  now  and  then 
useful,  but  ordinarily  a  tangent  to  a  curve  can  be  drawn 
with  all  necessary  accuracy,  in  the  first  example,  by  plac- 


LINES    AND    SURFACES    CLASSIFIED  97 

ing  a  straight  edge  to  touch  the  point  and  the  curve,  and  in 
the  second  example  the  point  of  tangency  can  be  estimated 
well  enough  for  all  practical  purposes,  in  those  cases  where 
the  curve  does  not  closely  approach  the  straight  line  tan- 
gent in  direction. 

98.  A  tangent  plane  to  a  surface  is  the  limit  approached 
by  a  secant  plane  when  three  points  of  the  curve  of  inter- 
section of  the  plane  with  the  surface,  which  do  not  lie  in 
the  same  straight  line,  become  coincident. 

A  tangent  plane  may  otherwise  he  defined  as  such  that, 
if  any  other  and  secant  plane  be  passed  to  cut  the  surface, 
it  will  cut  a  line  from  the  surface  and  a  line  from  the  tan- 
gent plane  which  are  tangent  to  each  other,  hence  the  tan- 
gent plane  at  a  point  is  the  locus  of  the  tangents  to  plane 
curves  which  may  be  drawn  on  the  surface  through  the 
point. 

For  this  we  may  derive  a  general  rule  for  passing  a 
plane  tangent  to  a  surface  at  a  given  point. 

Rule:— Through  the  given  point  conceive  two  secant  planes 
to  be  passed  to  cut  the  surface  in  at  least  two  plane  curves. 
Draw  tangents  to  these  at  the  given  point  and  the  plane  of 
the  tangents  will  he  the  plane  rectuired. 

A  right  line  is  normal  to  a  surface,  at  a  given  point, 
when  it  is  perpendicular  to  the  tangent  plane  at  that  point. 

A  plane  is  normal  to  a  surface  at  a  given  point,  when  it 
contains  a  line  normal  to  the  surface  at  the  point.  Hence, 
while  there  can  be  but  one  normal  line,  there  can  be  an  in- 
finite number  of  normal  planes  at  a  given  point  on  a  sur- 
face. 


CHAPTER   V. 


SINGLE  CURVED   SURFACES. 

99.  THEOREM    XL 

A  plane  which  contains  two  consecutive  rectilinear 
elements  of  a  single  curved  surface  will  be  tangent  to  the 
surface  throughout  these  elements,*  and  conversely,  if  a 
plane  is  tangent  to  a  single  curved  surface,  it  will  contain 
at  least  two  consecutive  rectilinear  elements  of  the  surface. 

Proof:— For,  if  through  any  point  of  one  element,  a 
secant  plane  is  passed,  it  will  cut  a  curve  from  the 
surface  having  consecutive  points  common  to  both  ele- 
ments. A  line  connecting  these  points  will  (by  Sec.  93) 
lie  in  the  plane  of  the  curve  of  the  section  and  (by  Sec. 
98)  in  the  tangent  plane.  Hence  the  given  plane  will  be 
tangent  at  the  assumed  point  or  all  along  the  element 
which  is  the  locus  of  the  assumed  points. 

100.  A  cylinder  is  defined  as  a  single  curved  surface 
generated  by  a  line  which  moves  so  that  its  successive 
positions  are  parallel,  and  touch  a  curve.  In  the  most 
common  interpretation,  the  curve  is  a  plane  curve. 


*  While   the  words,   'these  elements'  are  used,  It  Is  understood   that   the 
common  conception  of  '  an  element  of  tangency '  prevails. 


SINGLE    CURVED     SURFACES  99 

The  curve  of  intersection  with  a  cylinder,  of  a  plane 
perpendicular  to  the  generatrix  or  to  the  axis  of  the 
cylinder,  is  that  which  gives  the  cylinder  it  characteristic 
name  as  elliptical,  hyperbolic,  parabolic  cylinder,  etc.* 

A  cylinder  may  also  be  called  an  oblique  cylinder, 
when  considered  merely  in  reference  to  a  section  oblique 
to  the  generatrix,  and  then  the  name  of  the  cylinder  is 
commonly  derived  frctoi  the  shape  of  the  oblique  section. 

Any  secant  plane  of  a  cylinder,  making  a  greater  angle 
than  zero  degrees  with  the  elements,  may  be  considered  as 
a  base  of  the  cylinder  and  may  also  give  it  its  characteristic 
name.  It  is  customary  to  call  the  curves  of  section,  of  two 
parallel  secant  planes,  *the  bases'  of  the  cylinder. 

A  right  cylinder  is  one  whose  base  or  bases  are  perpen- 
dicular to  the  axis  or  elements. 

A  right  cylinder  with  a  circular  base  is  also  called  a 
cylinder  of  revolution  since  it  may  be  generated  by  one 
line  revolving  about  another  to  which  it  is  parallel. 

Let  Fig.  51,  page  100,  be  a  cylinder  with  circular  base 
in  H  and  another  base  parallel  to  H  with  elements  oblique 
to  the  plane  of  the  base.  This  cylinder  is  called  an  el- 
liptical cylinder  according  to  the  generic  definition  just 
given,  since  a  section  perpendicular  to  the  axis  would  be 
an  ellipse.  It  may  be  called  an  oblique  circular  cylinder, 
when  referred  to  its  oblique  bases. 

lOl.    To  assume  a  point  on  any   single  curved  surface, 

assume  one  projection  and  draw  the  projection  of  an  ele- 
ment through  the  point.    The  points  in  which  this  element 


*  The  axis  goes  through  the  center,  focus  or  other  characteristic  point  of  the 
curve  and  Is  parallel  to  the  generatrix.  Such  a  point  may  be  a  point  of  changing 
curvature  of  the  curve  of  tangents,  the  cusp  In  a  cuspidal  curve,  etc. 


100 


DESCRIPTIVE    GEOMETRY 

^1 


Figure  51. 


cuts  the  curve  or  curves  of  the  bases  give  points  in  the 
corresponding  projection  of  the  element;  and  the  cor- 
responding projection,  of  the  required  point  lies  on  the  cor- 
responding projection  of  the  element  through  the  point. 
An  element  of  a  single  curved  surface  may  be  assumed  in 
like  manner. 

For  example, let  p'  be  assumed  on  the  cylinder  in  Fig. 
51.  Draw  the  V  projection  of  an  element  CD  through  the 
point  p' .  Get  the  H  projection  of  this  element,  and 
through  p'  erect  a  perpendicular  to  the  G.L.  to  cut  the  H 
projection  of  the  element  CD  in  the  point  p  which  is  the 
corresponding  projection  of  the  point  required. 

Note  that  p'  would  stand  for  two  points  upon  the  sur- 
face of  the  cylinder,  for  through  the  point  P  a  perpendic- 
ular can  be  drawn  to  the  V  plane  and  it  would  pierce  the 
surface  in  two  points,  p^  being  the  other  H  projection  of 
the  point  corresponding  to  p ' . 


SINGLE  CURVED  SURFACES  101 

102.  To  draw  a  taagent  plane  to  a  cylinder  through  any 
point  on  the  surface. 

Analysis: — The  required  plane  will  contain  the  tangents 
to  curves  of  section  of  the  surface^  hut  an  element  of  contact 
may  he  taken  as  one  line  in  the  tangent  plane,  a  tangent  to 
the  curve  of  intersection  of  the  surface,  with  a  coordinate 
plane,  may  he  taken  as  the  other  line,  hut  the  latter  is  a 
trace  of  the  plane  since  it  lies  in  the  coordinate  plane. 
Hence,  draw  an  element  of  the  surface  through  the  given 
point  and  where  this  element  cuts  the  coordinate  plane,  in 
the  curve  of  the  hase,  draiv  a  tangent  to  the  hase.  The  plane 
of  these  two  lines  is  the  required  plane. 

If  the  plane  of  the  base  does  not  lie  in  a  coordinate 
plane,  draw  a  tangent  to  the  curve  of  the  base  at  the  point 
of  intersection,  with  it,  of  the  element  of  contact.  This, 
and  the  element  of  contact,  are  two  lines  of  the  required 
plane.  The  traces  of  the  latter  go  through  the  corres- 
ponding traces  of  these  lines. 

Construction — Let  the  cylinder  be  given  as  in  Fig.  51 
and  the  given  point  P.  Draw  the  element  DC  through  P 
and  at  c,  the  H  trace  of  DC,  draw  a  tangent  to  the  curve  of 
the  base.  It  is  the  H  trace  of  the  required  plane.  With 
the  H  trace  and  P  as  a  point  in  the  plane,  find  the  V  trace 
by  means  of  a  horizontal  of  the  plane. 

If  the  axis  of  the  cylinder  is  perpendicular  to  one  co- 
ordinate plane,  the  trace  of  the  tangent  plane  upon  the  cor- 
responding plane  of  projection  would  be  perpendicular  to 
the  G.L. 

If  the  axis  of  the  cylinder  is  parallel  and  its  hases  per- 
pendicular to  hoth  coordinate  planes,  it  is  necessary  to  use 
an  end  projection  to  get  the  points  of  intersection  of  the 
elements  with  the  curves  of  the  bases. 


102  DESCRIPTIVE    GEOMETRY 

103.    To  pass  a  plane  tangent   to  a  cylinder  through  a 
point  outside. 

Analysis:—//  an  auxiliary  line  is  drawn  through  the 
given  pointy  parallel  to  the  elements  of  the  cylinder  ^  this  line 
will  lie  in  the  tangent  plane  through  the  point.  The  plane 
of  this  line  and  the  element  of  contact  of  the  tangent  plane 
will  he  the  required  plane.  But  the  secant  plane  of  the  base 
will  cut  the  tangent  plane  in  a  line,  which  is  tangent  to 
the  curve  of  the  hase,  and  the  secant  plane  intersects  the 
auxiliary  line  drawn  through  the  given  point  parallel  to  the 
elements  of  the  surface.  Therefore,  find  the  intersection  of 
the  auxiliary  line  with  the  plane  of  the  base,  and  draw 
through  this  point  a  line  tangent  to  the  curve  of  the  base. 
The  plane  of  these  two  lines  is  the  required  plane.  There 
can,  in  general,  be  two  such  planes  as  two  tangents  can,  in 
general,  be  drawn  to  the  curve  of  the  base  from  a  given 
point  outside. 

Construction:— Let  the  cylinder  be  given  as  in  Fig.  52 
in  which  the  plane  of  a  base  is  in  V  and  its  center  in  the 
G.L.    Let  P  be  the  given  point. 

Draw  an  auxiliary  line  PQ  through  the  point  P  inter- 
secting the  plane  of  the  base  in  Q.  Q  is  one  point  in  a 
tangent  to  the  curve  of  the  base,  the  point  of  tangency 
being  C.  QC  is  a  line  of  the  required  plane  and  since  it 
lies  in  Y  also,  is  the  V  trace  of  the  required  plane  T.  CD, 
an  element  of  the  cylinder  through  C,  is  another  line  of 
the  tangent  plane.  Using  the  V  trace  and  P,  a  point  in 
the  plane,  a  vertical  of  the  plane  locates  a  point  in  the  H 
trace. 

In  this  problem  also  a  second  tangent  plane  R  may  be 
drawn. 


SINGLE  CURVED  SUP-FACES 


103 


Figure  52. 


104.    To  draw  a  tangent  plane  to  a  cylinder  and  parallel 
to  a  line  outside  the  cylinder. 

Analysis: — The  required  tangent  plane  must  contain  a 
line  parallel  to  the  given  line,  and  similarly,  if  a  plane  is 
passed  through  the  given  line  and  parallel  to  the  elements  of 
the  cylinder  it  will  he  parallel  to  the  required  plane.  Hence 
draw  a  line  parallel  to  the  elements  of  the  cylinder  to  inter- 
sect the  given  line.  The  plane  of  these  two  lines  is  parallel 
to  the  required  plane. 

To  draw  a  parallel  plane  tangent  to  the  cylinder,  pro- 
duce the  elements  of  the  cylinder,  if  necessary,  to  intersect  a 
coordinate  plane,  giving  the  curve  of  intersection  with  the 
latter.  Tangent  to  this  curve  and  parallel  to  the  similar 
trace  of  the  auxiliary  plane,  draw  the  trace  of  the  required 
plane,  from  which  the  corresponding  trace  may  he  ohtained. 

If  the  axis  of  the  cylinder  is  parallel  to  the  G.L.  and 
the  bases  perpendicular  to  it,  an  end  plane  will  have  to  be 


104 


DESCRIPTIVE    GEOMETRY 


used  to  get  the  traces  of  the   auxiliary  and  the  parallel 
tangent  planes. 


Figure    53 


Construction: — Let  the  cylinder  be  given  as  in  Figure 
53,  with  bases  parallel  to  V  and  centers  of  bases  at  A  and 
B  respectively;  and  let  the  given  line  be  KM.  Through 
any  point  of  KM,  as  M,  draw  a  line  MNO,  parallel  to  the 
elements  of  the  cylinder.  Through  its  traces,  and  those  of 
the  line  KM,  draw  the  traces  of  their  plane  T.  Prolong 
the  elements  of  the  cylinder  and  the  axis  to  pierce  the  Y 
plane,  the  latter  in  the  point  C.  Since  the  base,  with 
center'  at  A,  is  parallel  to  V,  the  intersection  of  the 
cylinder  with  V  will  be  also  a  circle,  with  center  at  C. 
Tangent  to  the  circle  of  the  base  in  V,  draw  the  V  trace  of 
a  tangent  plane  U  parallel  to  T.    By  means  of  the  H  trace 


SINGLE  CURVED  SURFACES  106 

of  the  element  of  contact  of  this  tangent  plane,  draw  the  H 
trace  of  the  plane  U  parallel  to  that  of  the  plane  T.  Two 
tangent  planes  can  be  drawn  to  fulfill  the  conditions. 

A  plane  cannot  be  drawn  tangent  to  a  cylinder,  and  to 
contain  a  given  line  outside,  unless  that  line,  prolonged,  is 
tangent  to  the  surface  at  a  finite  or  infinitely  distant  point. 

In  locating  the  traces  of  tangent  planes,  and  in  finding 
the  second  trace  when  one  trace  has  been  found,  the 
several  methods  may  be  used  which  have  already  been 
studied  in  the  fundamental  problems.  (1.)  A  point  in 
the  plane  can  be  taken,  and  a  horizontal,  a  vertical,  or  any 
other  auxiliary  line  drawn  through  the  point  to  lie  in  the 
plane.  (2.)  Any  line,  known  to  be  in  the  required  plane, 
can,  by  its  traces,  locate  the  trace  of  the  desired  plane. 

If  the  traces  of  the  required  plane  do  not  intersect  the 
G.  L.  within  the  limits  of  the  drawing,  it  will  be  generally 
necessary  to  find  the  traces  of  two  lines  in  the  plane,  un- 
less the  required  plane  is  parallel  to  some  other  given  or 
derived  plane.  Otherwise,  the  trace  of  only  one  auxiliary 
line  need  be  obtained. 

If  the  axis  of  the  cylinder  is  parallel  to  a  coordinate 
plane,  the  trace  of  a  tangent  plane,  upon  that  coordinate 
plane,  will  be  parallel  to  that  projection  of  the  elements  of 
the  cylinder. 

105.    To  pass  a  plane  normal  to  a  cylinder  through  a 
point  on  the  surface. 

Anelysis:— The  normal  plane  will  be  perpendicular  to 
the  tangent  plane  at  the  point,  i,e.,  contain  a  perpendicular 
to  the  tangent  plane  through  the  given  point.  Hence  find 
the    tangent  plane,   fhy    Section  102),  and,   through   the 


106  DESCRIPTIVE     GEOMETRY 

latter,  draw  a  perpendicular  to  this  plane.    Any  plane,  to 
contain  this  line,  will  satisfy  the  conditions. 

106.  To  pass  a  plane   normal  to  a  cylinder   through  a 
point  outside  the  cylinder. 

Analysis: — The  normal  plane  will  he  perpendicular  to 
the  tangent  plane  and  contain  the  point,  i.e.,  contain  a  per- 
pendicular to  the  tangent  plane  and  passing  through  the 
point.  Hence,  find  the  tangent  plane,  to  the  cylinder, 
through  the  given  point,  fhy  Section  103),  and  then  draw  a 
perpendicular  to  this  plane,  through  the  point.  Any  plane 
passed  through  this  perpendicular,  will  satisfy  the  condi- 
tions 

107.  To  pass  a  plane  normal  to  a  cylinder  and  parallel 
to  a  given  line. 

Analysis:— T/^e  normal  plane  will  he  perpendicular  to 
the  tangent  plane,  which,  in  turn,  is  parallel  to  the  given 
line.  Hence,  find  the  tangent  plane  to  the  cylinder,  parallel 
to  the  given  line,  fhy  Section  104. J  At  any  point  of  the 
tangent  plane,  erect  a  perpendicular,  and,  through,  any 
point  of  this  perpendicular,  draw  a  line  parallel  to  the  given 
line.    The  plane  of  these  two  lines  is  the  required  plane. 

108.  To  draw  a  plane  normal  to  a  cylinder,  through  a 
given  point  on  the  cylinder,  and  parallel  to  a  given  line. 

Analysis: — The  normal  plane  will  he  perpendicular  to 
that  tangent  plane  which,  in  turn,  is  parallel  to  the  given 
line.  Hence,  find  the  tangent  plane  to  the  cylinder  which 
is  parallel  to  the  given  line.  Through  the  given  point  on 
the  cylinder,  draw  a  perpendicular  to  this  tangent  plane, 
then  through  any  point  on  this  perpendicular,  draw  a  line 


SINGLE  CURVED  SURFACES  107 

parallel  to  the  given  line.  The  plane  of  these  two  lines  is 
the  required  plane. 

109.  The  elements  of  contour  of  a  solid  are  determined 
by  the  lines  of  tangeney  of  planes  respectively  perpendic- 
ular to  the  coordinate  planes,  and  these  planes  go  to  make 
up  the  projecting  cylinder  of  the  solid,  on  the  respective 
coordinate  planes.  In  the  case  of  cylinders  and  cones,  the 
contour  is  partly  made  up  of  the  projections  of  elements. 

It  can  be  seen  that  the  contour  elements  of  the  vertical 
projection,  will  not  be  identical  with  those  of  the  H  project- 
ion and,  therefore,  each  set  of  elements  of  contour  has  its 
corresponding  projections  which  do  not  coincide  with 
that  of  the  other. 

110.  The  curve  of  intersection  of  a  plane  and  a  surface 

is  found  by  plotting  the  points  of  intersection  of  lines, 
lying  respectively  in  each.  And,  for  this  purpose,  the 
following  general  rule  is  given  to  cover  all  kinds  of 
surfaces : 

Rule: — To  find  the  curve  of  intersection  of  a  plane  and  a 
surface,  pass  a  series  of  auxiliary  secant  planes,  so  chosen 
us  to  cut  straight  line  elements,  circular  section  or  such  lines 
from  the  solid  as  may  easily  he  drawn  in  projection;  these 
recant  planes  will  cut  right  lines  from  the  intersecting  plane 
surface,  the  intersections,  of  which,  with  the  elements  or  lines 
cut  from  the  surface  by  the  secant  planes,  will  give  points  in 
the  curve  of  the  section. 

In  single  curved  surfaces,  and  warped  surfaces,  it  is 
always  posssible  to  cut  out  straight  line  elements,  although 
these  may  not  in  all  cases  be  the  most  convenient  kind  of 
lines  to  cut. 


108  DESCRIPTIVE     GEOMETRY 

Rule:— To  find  the  piercing  point  of  a  line  with  a 
surface,  pass  any  convenient  plane  through  the  line  to  cut 
the  surface,  Find  the  curve  of  intersection  of  this  plane 
and  the  surface.  The  required  piercing  point  will  lie  at 
the  intersection  of  the  given  line  and  the  curve  cut  from  the 
surface.  The  line  may  pierce  the  surface  in  more  than 
one  point. 

111.    The  development  of  a  surface  of  single  curvature  is 

the  area  of  contact  of  the  surface,  when  it  is  rolled  at  one 
of  its  elements  upon  a  tangent  plane  to  that  element  until 
the  planes  of  the  consective  elements  come  successively 
into  contact  with  the  tangent  plane.  Only  surfaces  of 
single  curvature  and  polyhedrons  can  be  developed,  in  the 
true  sense  of  the  word,  although  any  surface  is  capable  of  a 
more  or  less  close  approximation  to  development,  when  it 
has  to  be  constructed  in  practical  work.  An  entire  surface 
may  be  developed,  or  any  portion  of  it,  lying  between  two 
chosen  elements  and  any  other  limiting  lines. 

112,  To  find  the  intersection  of  a  right  cylinder  v/ith 
a  plane  surface  oblique  to  its  axis  and  to  draw  a  tangent  to 
the  curve  of  intersection  at  any  point. 

Analysis: — The  intersection  is  a  plane  curve,  inter- 
secting all  the  elements  of  the  cylinder.  Hence,  pass  secant 
planes  to  cut  straight  lines  (elements)  from  the  cylinder, 
and  also  straight  lines  from  the  plane,  the  points  of  inter- 
section, respectively,  of  which  give  points  in  the  curve  of  the 
section. 

The  tangent  to  the  curve  of  intersection,  at  any  point 
by  Theorem  X,  Sec.  94,  will  be  projected  as  a  tangent  to 


SINGLE  CURVED  SURFACES 


109 


the  projection  of  the  curve,  but  it  is  most  accurately  drawn 
by  revolving  the  curve  of  intersection  about  the  trace  of  its 
plane  into  the  coordinate  plane  where,  by  the  properties  of 
the  curve,  it  may  be  more  accurately  drawn,  and  then  re- 
volved back  again,  noting  that  the  trace  of  the  tangent  is 
a  fixed  point  in  the  revolution. 


Figure  54. 


Construction:— Let    the  cylinder  be  a  right    circular 
cylinder,  with  base  in  H,   as  shown  in  Figure  54.     The 


110  DESCRIPTIVE     GEOMETRY 

center  of  the  base  is  at  A  and  let  the  plane  be  given  as  T. 
Auxiliary  planes,  to  cut  straight  line  elements  from  the 
cylinder,  are  parallel  to  the  axis  of  the  cylinder,  in 
this  case  perpendicular  to  H.  Let  S  be  such  a  plane, 
cutting  T  in  the  line  CD,  and  tangent  to  the  cylinder  along 
the  element  EF.  The  point  of  intersection,  O,  of  the  line 
CD  and  the  element  cut  from  the  cylinder,  is  a  point  in  the 
curve  of  intersection  of  the  plane  T  and  the  cylinder.  In 
a  similar  manner,  an  auxiliary  plane  R,  intersecting  T  in  a 
line  parallel  to  that  made  by  the  plane  S,  gives  P  and  Q, 
two  other  points  in  the  curve  of  intersection.  And  so  on, 
for  as  many  points  desired. 

Let  ^  be  a  chosen  point,  on  the  curve  of  intersection, 
at  which  to  draw  a  tangent.  Revolve  the  curve  of  inter- 
section into  V,  about  the  V  trace,  where  the  tangent  can  be 
drawn  more  accurately,  h\l'  is  the  revolved  position  of 
the  tangent.  The  point  L,  in  which  the  tangent  cuts  the 
V  trace,  is  a  fixed  point  in  the  revolution.  Revolve  the 
tangent  back  again  into  true  position  in  the  plane  T. 

Auxiliary  planes  could  have  been  passed  in  any  chosen 
direction,  for  example,  radially  through  the  axis  of  the 
cylinder,  or  parallel  to  each  other  in  any  other  direction. 
In  this  particular  case,  auxiliary  planes,  parallel  to  H, 
might  quite  as  conveniently  be  chosen;  they  would  have 
intersected  the  cylinder  in  circles,  readily  drawn,  and  their 
intersections  found,  with  the  horizontals  cut  from  the  plane 
T. 

113.  To  develop,  approximately,  the  right  circular 
cylinder  of  Sec.  112,  and  to  trace  on  it  the  curve  of  inter- 
section with  the  plane  T. 

Analysis:— From  definition  (Section  111)  the  develop- 


SINGLE  CURVED  SURFACES 


111 


ment  of  a  right  cylinder  with  parallel  bases,  is  hounded  hy 
the  rectified  curves  of  the  bases  and  the  elements  perpen- 
dicular to  and  connecting'  the  extremities  of  the  rectified 
bases. 

Hence,  choose  any  element,  and,  perpendicular  to  it  at 
its  extremities,  lay  off  the  rectified  curves  of  the  bases,  in 
any  convenient  manner,  i.e.,  for  example,  using  short  chord 
lengths  as  equal  to  the  arcs  which  subtend  them.  Connect 
the  last  divisions  on  the  rectified  curves  to  constitute  the 
element  consecutive  with  the  first  one  assumed.  The  curve 
of  intersection  of  a  secant  plane  will  be  a  curve  connecting 
the  proper  points  of  the  respective  elements  as  found  in  the 
development.  Hence,  lay  off  on  each  intersected  element 
in  the  development,  a  distance  from  the  rectified  curve  of  a 
base  that  the  point  of  intersection  of  the  element  with  the 
secant  plane  is  from  that  base;  draw  a  smooth  curve 
through  these  points. 

Construction: — Take  any  reference  line  as  the  upper 


F  I  QUBE    55. 


one  in  Figure  55,  and  lay  off  on  it  successively,  starting  at 
the  left,  and  going  toward  the  right,  the  chord  lengths 


112  DESCRIPTIVE    GEOMETRY 

into  which  the  base  may  have  been  divided  by  the 
auxiliary  planes  in  H  projection,  beginning  at  p  and  pro- 
ceeding counter  clockwise  around  the  base.  At  these 
divisions  erect  perpendiculars  to  the  reference  line  or  base, 
and,  making  them  as  long  as  the  elements  are  long,  obtain 
the  other  base.  Next,  divide  the  elements  of  the  develop- 
ment into  the  same  proportionate  parts  that  the  elements 
are  divided  in  projection  by  the  secant  plane.  A  smooth 
curve  connecting  these  points  will  give  the  developed 
curve  of  intersection. 

114.  To  develop  an  oblique  cylinder. 
Analysis:— To  develop  any  cylinder,  it  is  desirable  to 
obtain,  first,  a  right  section.  Then  rectify  this,  and  erect 
the  elements  perpendicular  to  it;  lay  off  on  these  the  respec- 
tive lengths  between  the  bases  measured  from  the  right 
section  as  a  reference  line. 

If  the  axis  of  the  cylinder  is  oblique  to  both  coordinate 
planes,  it  is  most  convenient  to  change  the  latter  so  that 
the  axis  is  parallel  to  one  of  them,  for  then,  one  projection 
of  the  curve  of  intersection  will  be  a  straight  line,  and  the 
construction  for  getting  the  other  projection  of  the  inter- 
section is  very  much  simplified. 

Construction:— Let  an  oblique  elliptical  cylinder  with 
parallel  bases,  and  axis  AB  oblique  to  H  and  Y,  be  given  as 
in  Figure  56.  Assume  any  plane  T,  conveniently  between 
the  bases  of  the  cylinder  to  be  perpendicular  to  the  axis. 
Its  traces  are  respectively  perpendicular  to  the  projections 
of  the  axis  of  the  cylinder.  To  find  the  curve  of  inter- 
section of  T  with  the  cylinder,  project  the  latter  upon  a  new 
Vi  plane  parallel  to  the  axis  cutting  H  in  Gi  .Li  .    The  new 


SINGLE  CURVED  SURFACES 


113 


FIGUKE     56. 


Vi  trace  of  T  will  be  perpendicular  to  the  Vi  projection  of 
the  axis  a\b\  .  Pass  planes  perpendicular  to  H  and  to  T, 
as  R  and  S  for  example,  to  cut  elements  CD,  EF  and  GI, 
JK  respectively  from  the  cylinder  as  shown.  The  V^  pro- 
jections of  the  piercing  points  of  the  elements  lie  in  the 
Vi  trace  of  T;  the  H  projections  lie  at  the  intersection  of 
perpendiculars  to  the  Gi.Lj.  through  these  points  and  the  H 
projections  of  the  respective  elements  cut  from  the  cylin- 
der. 


114 


DESCRIPTIVE    GEOMETRY 


To  get  the  true  curve  of  the  intersection,  revolve  it  in- 
to or  parallel  to  the  V  plane,  for  example  about  the^V  trace 
of  T  as  shown  above  TV  on  the  figure. 


Figure   57. 


Next,  rectify  the  curve  of  the  section  on  any  base  or 
reference  line,  as  in  Fig.  57,  by  laying  off  on  it  any  conven- 
ient chord  lengths  as  taken  from  the  rectified  curve  of  sec- 
tion and  erecting  perpendiculars  to  the  points  of  divis- 
ion. Lay  off  on  these  perpendiculars,  either  side  of  the 
rectified  curve  of  the  section,  distances  respectively  equal 
to  the  segments  of  the  elements  upon  either  side  of  the 
curve  of  section.  The  true  lengths  of  all  elements  are  pro- 
jected upon  the  Vi  plane  and  are  obtained  directly. 


SINGLE    CURVED     SURFACES  115 

116.  A  cone  is  defined  as  a  single  curved  surface  gen- 
erated by  the  motion  of  a  line  which  goes  through  a  fixed 
point  and  touches  a  curve.  There  is  nothing  which  limits 
the  curve  but  it  is  usual  to  consider  it  a  plane  curve. 

The  generic  or  characteristic  name  of  a  cone  is  derived 
from  the  perimeter  of  a  plane  section  which  is  so  taken 
that  a  line  through  the  apex  of  the  cone,  and  perpendicular 
to  the  plane  section,  goes  through  its  center,  focus,  or  other 
characteristic  point  of  the  curve.*  Some  cones  so  defined 
are  elliptical,  hyperbolic,  parabolic,  etc.  The  axis  of  a 
cone  is  a  line  which  goes  through  the  apex  and  pierces  a 
plane  section,  arbitrarily  chosen  as  the  base,  in  its  center, 
focus,  or  other  characteristic  point.  The  axis  of  a  generic, 
or  right  cone,  is  such  a  line  which  is,  further,  perpendicu- 
lar to  the  plane  of  the  base. 

A  cone  may  he  at  the  same  time  a  right  cone  and  an  ob- 
lique cone.  The  definition  just  given  for  a  generic  cone 
makes  every  cone  some  form  of  right  cone.  Any  plane 
section,  however,  oblique  to  the  generic  axis  may  also  give 
the  cone  a  name.  A  simple  illustration  will  suffice:  A 
right  circular  cone,  if  cut  by  a  plane  oblique  to  its  axis, 
may  also  be  called  an  oblique  elliptical  cone.  A  cone  is 
generally  defined  in  terms  of  the  base  section  whether  it 
be  oblique  to  the  axis  or  perpendicular  to  it. 

A  cone  of  revolution  is  a  right  circular  cone,  because 
it  can  be  generated  by  the  revolution  of  a  line  about 
another  line  which  it  intersects. 

The  generating  line  or  generatrix  of  any  cone  is  not  limit- 

*The  author  believes  this  to  be  an  uncommon  definition  and  therefore  re- 
quires some  fuller  explanation.  The  'characteristic  point'  may  mean  a  point  oi 
Inflection  of  the  curve  of  section  if  It  has  sinuous  like  characteristics,  a  cusp, 
or  a  vertex,  etc. 


116  DESCRIPTIVE    GEOMETRY 

ed  by  the  apex  of  the  cone,  therefore,  a  line  always  gen- 
erates two  parts,  each  on  opposite  sides  of  the  fixed  point. 
These  parts  are  called  nappes,  as  the  right  and  left  nappe 
respectively,  or  the  upper  and  lower  nappe. 

116.  To  assume  a  point  on  a  cone,  assume  either  pro- 
jection of  the  point,  and  through  it,  draw  the  projection  of 
an  element,  which  is  a  line  of  the  surface  passing  through 
the  apex.  Note  where  it  cuts  the  curve  of  the  base,  with 
this  point  construct  the  other  projection  of  the  element. 
The  other  projection  of  the  assumed  point  will  be  on  the 
corresponding  projection  of  the  element.  Each  assumed 
point  lies,  in  general,  on  the  projection  of  two  elements. 
Either  one  can  be  taken  at  will.  An  element  of  the  cone 
may  be  assumed  in  a  similar  manner. 

117.  To  draw  a  tangent  plans  to  a  cone  at  any  point  on 
the  surface. 

Analysis:— T/^e  plane  will  contain  the  tangent  to  two 
curves  of  section  of  the  surface  through  the  point,  hut  an 
element  of  contact,  may  he  taken  as  one  line  of  the  tangent 
plane,  a  tangent  to  the  curve  of  intersection  of  the  surface 
with  a  coordinate  plane  mag  he  taken  as  the  other  line, 
hut  the  latter  is  also  a  trace  of  the  plane,  since  it  lies  in 
the  coordinate  plane.  Hence,  draw  an  element  of  the  surface 
through  the  given  point  and  where  this  cuts  the  coordinate 
plane  in  the  curve  of  a  hase  in  that  plane,  draw  a  tangent  to 
the  hase.    The  plane  of  these  two  lines  is  the  required  plane. 

If  the  plane  of  the  hase  does  not  lie  in  a  coordinate 
plane,  draw  a  tangent  to  the  curve  of  the  hase  at  the  point 
of  intersection  with  it  of  the  element  of  contact.    This  and 


SINGLE  CURVED  SURFACES 


117 


the  element  of  contact  are  lines  of  the  required  plane.  The 
traces  of  the  tangent  plane  go  through  the  respective  traces 
of  these  lines. 


Figure  58. 

Construction:— Let  an  oblique  elliptical  cone  be  given 
as  in  figure  58  with  its  elliptical  base  in  H  and  apex  at  0, 
and  the  point  P  by  its  horizontal  projection.  Find  p'  by 
drawing  the  element  o  ^  to  cut  the  curve  of  the  base  in  r. 
r'  is  in  the  G.  L.  and  p'  is  on  r'o'.  The  required  tangent 
plane  contains  the  element  O  P  R  and  since  the  base  of  the 
cone  is  in  H,  a  tangent  to  the  curve  of  the  base  at  r,  the 
foot  of  the  element,  is  the  H  trace  of  the  required  plane. 
The  y  trace  is  obtained  by  using  a  horizontal  of  the  plane 
through  O  as  shown. 

118.    To  pass  a  plane  tangent  to  a  cone  through  a  point 
outside. 

Analysis: — If  an  auxiliary  line  is  drawn  through  the 
given  point  and  the  apex  of  the  cone^  it  will  he  a  line  of  the 


118 


DESCEIPTIVE    GEOMETRY 


tangent  plane,  for  all  tangent  planes  to  the  cone  go  through 
the  apex.  Another  line  of  the  tangent  plane  is  a  line  lying 
in  any  secant  plane,  intersecting  the  auxiliary  line  men- 
tioned and  tangent  to  the  curve  of  intersection  with  the  cone 
of  the  secant  plane.  If  the  base  of  the  cone  is  in  a  coordi- 
nate plane,  this  second  line  would  be  that  passing  through 
the  trace  of  the  auxiliary  line  and  tangent  to  the  curve  of 
the  base  in  the  coordinate  plane;  it  would  furthermore  be  a 
trace  of  the  tangent  plane.  The  other  trace  may  be  obtained 
by  any  of  the  usual  methods. 


Construction:— Let  an  oblique  elliptical  cone  be  given 
as  in  Fig.  59  with  base  in  H  and  apex  at  O  as  shown.  Let 
the  point  outside  be  R.  Draw  a  line  O  E  through  the  apex 
of  the  cone  to  pierce  H,  the  plane  of  the  base  in  W. 
Through  W  draw  a  tangent  to  the  curve  of  the  base.    This 


SINGLE  CURVED  SURFACES  U9 

will  be  at  once  a  line  of  the  required  plane  R  and  the  H 
trace  of  the  plane.  By  means  of  a  horizontal  OB  of  the  re- 
quired plane  the  V  trace  is  obtained. 

By  similar  construction  another  tangent  plane  Q  may 
be  found  to  fulfill  the  conditions. 

119.  To  draw  a  plane  tangent  to  a  cone  and  parallel  to  a 
line  outside. 

Analysis:— T/te  required  tangent  plane  must  contain  a 
line  parallel  to  the  given  line.  Such  a  line  can  he  drawn 
through  the  apex  of  the  cone,  A  plane  tangent  to  the  cone 
and  containing  this  line  would  fulfill  the  conditions.  Hence ^ 
draw  a  line  through  the  apex  parallel  to  the  given  line,  and 
through  its  piercing  point  with  the  plane  of  the  base,  draw 
a  tangent  to  the  curve  of  the  base.  This  line,  and  the  one 
through  the  apex,  are  lines  of  the  required  plane.  Since  two 
or  more  tangents  can  he  drawn  from  a  point  to  a  curve, 
there  can  he  two  or  more  tangent  planes. 

The  construction  is  left  to  the  student. 

A  plane  cannot  be  drawn  tangent  to  a  cone  and  to  con- 
tain a  given  line  outside  unless  the  line,  prolonged  if  neces- 
sary, would  be  tangent  to  its  surface. 

If  the  axis  of  the  cone  is  parallel  to  the  G.  L.,  the 
plane  of  the  base  will  be  an  end  plane  and  the  construction 
requires  to  be  modified  to  suit. 

120.  *  To  pass  a  plane  normal  to  a  cone  through  a  point 
on  the  surface. 

Analysis:— T/te  normal  plane  will  he  perepndicular  to 
the  tangent  plane  at  the  point,  i.e.,  contain  a  perpendicular 

*The  analysis  of  the  f ollowlne  three  problems  are  the  same  as  the  correspond- 
Ins  ones  for  the  cylinder. 


120  DESCRIPTIVE     GEOMETRY 

to  the  tangent  plane  through  the  given  point.  Hence,  find 
the  tangent  plane  at  the  point,  (hy  Section  117),  and 
through  the  latter  draw  a  perpendicular  to  this  plane.  Any 
plane  to  contain  this  line  will  satisfy  the  conditions. 

121.  To  pass  a  plane  normal  to  a  cone  through  a  point 
outside  the  cone. 

Analysis: — The  normal  plane  will  be  perpendicular  to 
the  tangent  plane  and  contain  the  point,  i.e.,  contain  a  per- 
pendicular to  the  tangent  plane  and  passing  through  the 
point.  Hence,  find  the  tangent  plane,  Cby  Section  118),  and 
draw  a  perpendicular  to  the  plane  through  the  point.  Any 
plane  passed  through  this  perpendicular,  will  satisfy  the 
conditions. 

122.  To  pass  a  plane  normal  to  a  cone  and  parallel  to 
a  given  line. 

Analysis:— T/^e  normal  plane  will  he  perpendicular  to 
the  tangent  plane  which  latter  is  also  parallel  to  the  given 
line.  Hence,  find  the  tangent  plane  to  the  cone  and  parallel 
to  the  given  line,  (hy  section  119.)  At  any  point  of  the 
tangent  plane  erect  a  perpendicular,  and  through  any  point 
of  this  perpendicular  draw  a  line  parallel  to  the  given  line. 
The  plane  of  these  two  lines  is  the  required  plane. 

123.  To  find  the  curve  of  intersection  of  any  cone  with 
~vl  plane  oblique  to  the  axis. 

Analysis: — The  intersection  will  he  a  plane  curve,  inter- 
secting all  the  elements  of  the  cone.  Hence,  pass  auxiliary 
secant  planes  to  cut  straight  lines,  elements,  from  the  cone 
and  lines  from  the  given  plane;  the  points  of  intersection, 
respectively ,  give  points  in  the  curve  of  the  section. 


SINGLE  CUKVED  SURFACES  121 

Construction:— Let  an  oblique  elliptical  cone  be  given 
as  in  Figure  60,  with  elliptical  base  in  H,  and  cut  by  a 
plane  T.  Pass  a  series  of  secant  planes,  one  of  which  is  S, 
through  the  apex  of  the  cone,  assuming  at  will  the  H 
traces  of  these  planes,  as  SH.  The  V  traces  can  be  found 
by  means  of  horizontals,  as  shown  in  the  case  of  plane 
S.     Find  the  lines  of  intersection,  respectively,  of  these 


Figure   60. 


auxiliary  secant  planes  with  the  plane  T,  for  example, 
CD  is  the  line  of  intersection  of  the  plane  S  with  the  plane 


122  DESCRIPTIVE    GEOMETRY 

T.  The  elements  cut  from  the  cone  by  the  plane  S,  are 
EG  and  FO,  respectively.  Where  e' o'  cuts  c' d'  is  a 
point  in  the  curve  of  intersection,  and,  where  co  cuts  cd,  is 
its  corresponding  projection.  Similarly,  where  f'o'  cuts 
c' d\  is  another  point  in  the  curve  of  intersection  and 
where  fo  cuts  cd,  is  its  corresponding  projection,  and  so 
on  for  as  many  points  as  desired. 

Either  the  V  or  the  H  traces  of  the  auxiliary  secant 
planes  can  be  assumed  in  this  problem  in  any  direction 
which  seems  convenient,  the  corresponding  traces  to  be 
determined  from  them. 

The  well  known  conies  are  derived  by  the  intersection 
of  a  plane  and  a  right  circular  cone,  or  cone  of  revolution 
as  was  discussed  in  Sections  86  and  115. 

124.  To  develop  a  right  circular  cone  approximately.* 
Analysis:— 7^  a  right  circular  cone,  all  the  elements  are 
of  equal  length.  Hence,  since  the  apex  is  a  point  common  to 
all  the  elements,  the  development  will  consist  of  the  sector 
of  a  circle,  the  distance  between  the  sides  of  which,  is 
determined  by  the  length  of  the  rectified  curve  of  the  base 
subtending  the  angle  between  the  sides  and  laid  out  with  the 
apex  as  center. 

Therefore,  with  a  radius  equal  to  the  true  length  of  an 
element,  draw  the  arc  of  a  circle  and  lay  off  on  it  from 
any  convenient  point,  successively,  chord  lengths  taken 
equal  to  short  arc  lengths  into  which  the  base  may  be 
divided.  Connect  the  first  and  last  point  of  the  chords 
with  the  center  or  apex  and  this  will  be  the  approximate 
development. 

*  Since  the  circle  of  the  base  caunot,  by  geometry,  be  accurately  rectified. 


SINGLE  CURVED  SURFACES 


123 


Figure   61 


Construction.— Let  a  right  circular  cone  be  given,  as  in 
Figure  61,  with  base  parallel  to  H  and  apex  in  0.  Divide 
the  curve  of  the  base,  for  convenience,  into  a  number  of 
equal  divisions,  taking  the  chords,  connecting  these,  as 
equal  to  the  respective  arc  lengths.  Choose  a  center  O 
and,  with  a  radius  equal  to  the  length  of  the  elements  of 
the  cone,  which  is  shown  in  true  projection  by  those 
elements  which  are  parallel  to  V  as  OP  and  OQ,  describe 
an  arc  of  a  circle  PQP  and,  on  this  circle,  lay  off  succes- 
sively the  chord  lengths  into  which  the  circle  of  the  plan 
had  been  divided.  Connect  the  first  and  last  division  with 
the  center  O  and  this  is  the  approximate  development. 

Any  other  than  a  right  circular  cone  will  have  to  be 
developed  by  a  different  method  than  the  foregoing. 


124  DESCRIPTIVE     GEOMETRY 

125.    The  development  of  any  cone  in  general,  may  be 
effected  as  follows: 

Analysis:— Divit^e  the  curve  of  the  base,  or  intersection 
of  the  cone  with  a  plane,  into  a  large  number  of  parts 
taking  the  chord  lengths  as  equal  to  the  arcs  subtending 
them.  Draw  the  corresponding  elements  of  the  cone  through 
these  divisions.  Then,  taking  an  apex  or  center  anywhere, 
assume  a  line  through  it,  equal  in  length  to  an  element  of 
the  cone,  next,  with  the  apex  as  center,  draio  an  arc  of  a 
circle  of  radius,  equal  to  the  true  length  of  the  next  adjoin- 
ing element,  and  with  center  at  the  extremity  of  the  first 
element,  and  radius  equal  to  the  true  chord  length  between 
the  two  elements,  describe  an  arc  of  a  circle  to  cut  the  first 
arc  in  a  point  which  will  lie  upon  the  developed  curve  of  the 
base.  With  centers,  respectively ,  as  before,  at  the  apex 
and  the  extremity  of  the  last  located  element,  and  radii 
equal,  respectively,  to  the  true  length  of  the  next  element, 
and  the  true  chord  length  between  the  elements,  find  the  next 
point  in  the  developed  curve  of  the  base  and  so  on  until  all 
the  points  of  the  base  selected  have  been  located.  The  first 
and  last  elements,  and  a  smooth  curve  through  the  points, 
plotted,  constitute  the  development. 

If  the  right  section  of  an  oblique  cone  is  a  circle  then 
the  following  special  method  for  development  may  also  be 
employed:  — 

Analysis:— Pas5  a  plane  perpendicular  to  the  axis.  It 
will  cut  the  cone  in  a  circle.  Develop  this  right  circular 
cone,  by  Section  124,  and  find  points  in  the  curve  of  the 
desired  base  by  laying  off  upon  the  developed  elements^ 
extended  if  necessary,  the  respective  rectified  intercepts  on 
those  elements  between  the  base  and  the  curve  of  intersection 


SINGLE  CURVED  SURFACES 


125 


with  the  auxiliary  plane,     A  smooth  curve  through  these 
points  will  give  the  curve  of  the  base. 

126.    To  develop  an  oblique  cone. 
Analysis: — The  analysis  is  the  same  as  that  given  for 
the  general  case  at  the  beginning  of  Section  125. 


F  IGURE    62. 

Construction— Let  an  oblique  cone  with  circular  base 
in  H  and  axis  in  an  end  plane  be  given  as  in  Figure  62. 
Divide  the  base  inH  into,  say,  sixteen  equal  parts.  For 
symmetry  in  the  development  choose  element  OA,  to  lay 
off  in  a  convenient  place,  obtaining  its  true  length,  by  re- 
volving the  element  parallel  to  V.  Find  the  true  lengths  of 
all  the  elements  by  similiar  revolution.  Next,  with  O,  of 
the  development,   as    center,    and    radius  equal  to  OB, 


126  DESCRIPTIVE    GEOMETRY 

describe  an  arc  of  a  circle,  then,  with  a 6  as  a  radius,  and 
center  at  A  of  the  development,  describe  an  intersecting 
arc  obtaining  the  point  B.  Similiarly,  obtain  all  the 
desired  points  in  the  curve  of  the  base.  The  elements 
OA  of  the  development  are  theoretically  consecutive 
elements  of  the  curve. 

127.  The  con  volutes  have  already  been  defined  broadly 
in  Sec.  87  but  to  repeat  for  the  sake  of  clearness,  they  are 
a  class  of  surfaces  generated  by  the  motion  of  a  line  which 
is  tangent  to  a  curve  of  double  curvature. 

Bearing  in  mind  the  discussion  in  Sec.  87,  that  when  a 
line  moves  tangent  to  a  curve  of  double  curvature  the 
elements  of  the  surface  generated  intersect  each  other  two 
and  two,  that  is,  two  elements  intersect  each  other  in  a 
point  of  the  curve  and  pass  through  two  other  consecutive 
points,  no  three  elements  passing  through  four  consecutive 
points,  and  it  can  be  seen  that  the  surface  is  developable, 
like  other  single  curved  surfaces. 

There  can  be  an  infinite  variety  of  convolute  surfaces 
depending  upon  the  form  of  the  curve  of  double  curvature 
or  directrix.  One  of  the  well  known  surfaces  of  this  char- 
acter, used  very  generally,  as  an  example  in  discussing  the 
properties  of  the  surface,  is  the  helical  convolute,  so 
named  because  it  is  generated  by  a  line  moving  so  that  it 
is  tangent  to  a  helix.* 

128.  The  helix  has  already  been  defined  as  the  curve 
traced  by  a  point  which  moves  at  a  uniform  rate  around  a 
cylinder,   and  at  the  same  time  at  a  uniform  rate  in  a 

*This  surface  is  also  called  a  developable  helicoid,  because  it  is  the  only  hell- 
coidal  surface  which  can  be  developed. 


SINGLE  CURVED  SURFACES 


127 


direction  parallel  to  the  axis  of  the  cylinder,  and,  since 
these  relations  are  directly  proportional  relations,  a  curve, 
plotted  between  them  in  coordinates,  will  be  a  straight, 
line,  so  that  the  helix  is  also  the  shortest  line  which  can  be 
drawn  on  a  cylinder,  between  two  points  that  lie  neither 
upon  the  same  right  section,  or  upon  the  same  right  line 
element. 


Figure    63. 


To  study  the  curve  in  detail,  see  Figure  63,  which 
shows  it  traced  on  a  circular  cylinder,  in  tnird  angle  pro- 
jection. Assume  a  point  to  be  at  O,  and  to  move  around  the 
cylinder  in  the  direction  of  the  arrow  through  equal  distan- 
ces, 1,  2, 3, 4,  etc.  Let  it  also  move  up  the  cylinder  through 
any  given  distance,  until,  after  it  has  completed  one  revo- 


128  DESCRIPTIVE    GEOMETRY 

lution,  it  reaches  a  position  P  directly  above  O.  The  dis- 
tance OP  is  known  as  the  pitch  of  the  curve.  Now,  as  both 
motions  are  uniform,  the  point  will  travel  to  1,  which  is  one- 
sixteenth  of  the  circumferential  distance,  in  the  same  time 
that  it  travels  one-sixteenth  of  the  distance  of  OP  toward 
P,  and  to  2,  which  is  one-eighth  of  the  circumferential 
distance,  as  it  goes  one-eighth  of  the  distance  OP  toward  P, 
and  so  on.  Hence  to  plot  the  curve,  divide  the  circumfer- 
ence of  the  base  into  any  convenient  number  of  equal 
parts,  and  the  pitch  into  the  same  number  of  equal  parts. 
By  noting  the  points  of  intersection  of  the  perpendiculars 
to  the  G.L.,  through  the  divisions  of  the  circumference  of 
the  base,  and  parallels  to  the  G.L.,  through  corresponding 
divisions  of  the  pitch,  points  of  the  curve  may  be  found. 

Certain  peculiarities  of  this  curve  deserve  to  be  noted. 
(1)  It  is  tangent  to  the  contour  elements  of  the  cylinder  at 
points  o'  and  8'  and  p' .  (2)  It  changes  curvature  at 
points  4'  and  12 \  midway  of  the  contour  elements.  (3)  A 
tangent  to  the  curve  at  4'  and  12'  shows,  by  its  angle  with 
the  projection  of  the  base  of  the  cylinder,  the  angular  pitch 
of  the  curve,  i.  e.,  the  ratio  of  the  linear  pitch  to  the  cir- 
cumferential distance.  (4)  The  curve  is  sharpest  at  o',  5' 
andp'  and  gradually  grows  straighter  until  at  4'  and  12' 
it  is  straight  for  a  very  short  distance.  (5)  It  is  sym- 
metrical, in  parts,  with  respect  to  the  projection  of  the  axis 
of  the  cylinder,  and  to  lines  perpendicular  to  the  projection 
of  the  axis,  so  that  the  curve  from  o'  to  4'  is  a  unit,  which 
is  repeated  throughout  the  path  of  motion  of  the  point. 

The  surface  of  the  helical  convolute  is  represented  by 
drawing  its  helical  directrix,  at  least  two  limiting  elements, 
audits  intersection  with  a  plane,  as  a  coordinate  plane. 


SINGLE    CURVED    SURFACES  129 

129.    To  represent  a  helical  convolute  with  its  axis  per- 
pendicular to  H. 

^Analysis:— T/^e  helical  directrix  will  appear  in  both  pro- 
jections as  described  in  Section  128,  The  V  projection  will 
he  limited  by  tangents  to  the  helical  directrix  at  the  points  of 
changing  curvature.  The  H  projection  of  the  surface  ivill 
be  as  follows:  The  tangents  to  the  helical  directrix  will  be 
projected  as  tangents  to  the  circle  of  the  H projection.  Since 
the  pitch  is  constant,  any  tangent  will  malce  a  constant  angle 
with  the  H plane  and  is  the  hypotenuse  of  a  right  angled  tri- 
angle, of  which  the  H  projection  of  the  tangent,  between  the 
curve  and  the  H plane,  is  one  side,  and  the  projecting  line  on 
the  H  plane,  of  any  point  of  the  tangent,  is  the  other  side. 
Moreover,  from  the  fact  that  the  tangent  has  the  pitch  of  the 
directrix,  at  the  point  of  tangency,  and  from  the  directly 
porportional  relation  in  the  motion  of  the  point,  in  generating 
the  curve,  the  H  projection  of  a  tangent  is  the  length  of  the 
developed,  or  unrolled  portion  of  the  directrix  or  circle  in  H 
projection.  Hence,  if  a  series  of  tangents  are  drawn  to  the 
directrix,  in  H projection,  and  are  made  respectively  equal  in 
length  to  the  circumferential  distance  between  the  H  trace  of 
the  directrix  and  the  respective  points  of  tangency,  the  ex- 
tremities of  the  tangents  will  lie  upon  an  involute  of  the 
circle.  The  intersection  of  the  helical  convolute,  by  the  H 
plane,  or  plane  perpendicular  to  the  axis,  is  therefore  the  in- 
volute of  a  circle. 

Construction:— Let  the  cylinder  and  the  helical  direc- 
trix be  given  as  in  Figure  64.  At  points  1,  2,  3,  etc.,  of 
the  V  projection  conceive  tangents  to  be  drawn  to  the 

♦Which  is  an  analysis  lor  special  conditions. 


130 


DESCRIPTIVE     GEOMETRY 


SINGLE    CURVED    SURFACES  131 

curve.  Since  the  traces  of  these  tangents  with  the  H 
plane  cannot  be  obtained  with  accuracy,  in  this  projection, 
derive  them  by  first  getting  the  corresponding  projection 
of  the  traces  as  follows:  O  is  the  intersection  of  the 
directrix  with  H.  Proceeding  from  O  counter  clockwise, 
draw  the  H  projections  of  a  series  of  tangents  to  the  base 
in  H,  and  make  the  length  of  each  tangent  equal  to  the 
rectified  arc  between  the  point  of  tangency  and  the  point 
O  giving  the  points  Q,  R,  S,  T,  U.,  etc.  Join  these  points 
with  a  smooth  curve  which  is  the  curve  of  intersection  of 
the  surface  with  the  H  plane,  and  the  latter  would  be 
called  its  base. 

Now  since  the  tangents  are  not  limited  in  length  by 
their  traces  and  points  of  tangency,  the  surface  will  be 
continuous  below  the  H  plane ;  a  portion  of  that  part  of  it 
which  is  below  is  shown  in  dotted  line. 

If  a  plane  parallel  to  H  were  to  intersect  the  surface 
at  a  distance  above  H  that  3  is  above  the  Gr.L.,  then  the 
curve  of  intersection  would  be  an  equal  curve  to  the  one  in 
H  but  it  would  have  a  cusp  at  3  instead  of  at  O. 

The  elements  are  the  lines  of  greatest  declivity  of  the 
surface  with  respect  to  any  plane  perpendicular  to  the 
axis,  and  the  surface  itself  is  one  of  equal  declivity  with 
such  a  plane,  in  this  case  with  the  H  plane. 

131.  To  assume  a  point  upon  the  surface  of  a  helical 
convolute,  and  to  pass  a  plane  tangent  to  the  surface  to 
contain  it. 

Analysis:— /SMce  a  point  lies  upon  an  element  of  the 
surface,  assume  either  projection  of  the  point,  and  draw  an 
element  through  it  hy  drawing  a  tangent  to  the  helical 


132  DESCRIPTIVE    GEOMETRY 

directrix.    The  corresponding  projection  of  the  point  will  lie 
upon  the  corresponding  projection  of  the  element. 

In  accordance  with  the  definition  of  a  tangent  plane  to 
a  single  curved  surface,  the  required  plane  will  he  that  of 
an  element  of  contact  through  the  given  point,  and  a  tangent 
to  the  curve  of  intersection  of  the  surface  with  any  plane. 
Hence,  draw  an  element  and  through  the  trace  of  that 
element  with  any  secant  plane,  draw  a  tangent  to  the  curve 
of  intersection  of  the  surface  with  that  secant  plane.  The 
plane  of  these  two  lines  is  the  required  plane.  If  the  secant 
plane  is  a  coordinate  plane,  then  the  tangent  to  the  curve  of 
intersection  with  the  secant  plane  is  at  once  a  trace  of  the 
required  plane. 

Construction: — Let  the  surface  be  given  as  in  Figure 
64.  Assume  a  point  a  on  the  surface.  Draw  an  element 
through  it  to  pierce  H  in  the  curve  s  t  u.  From  h ' ,  the  V 
projection  of  this  trace,  draw  a  tangent  to  the  V  projection 
of  the  helical  directrix  and  thus  obtain  a' , 

Through  h  draw  a  tangent  to  the  curve  s  t  u  and  this 
is  the  H  trace  of  the  required  tangent  plane.  With  the  H 
trace,  and  the  elements  BA  or  the  point  A  in  the  plane, 
find  the  V  trace. 

132.    To  find  the  curve  of    intersection  with  a  helical 
convolute  of  any  secant  plane  oblique  to  the  axis. 

Xn^Xysis:— According  to  the  rule  previously  given  for 
finding  the  curve  of  intersection  of  a  plane  with  any  single 
curved  surface,  a  series  of  auxiliary  secant  planes  may  he 
chosen  to  cut  lines  from  hoth.  The  most  convenient  planes, 
however,  will  prohahly  he  the  projecting  planes  of  the 
elements  of  the  convolute      Where  these  elements  pierce  the 


SINGLE    CURVED     SURFACES  133 

given  plane,  will  give  points  upon  the  curve  of  intersection. 
Hence,  hy  Section  57,  find  the  piercing  points  of  any  series 
of  elements  with  the  secant  plane,  and  join  them  hy  a  smooth 
curve. 

If  auxiliary  planes  are  passed  parallel  to  the  plane  of 
the  base  of  the  cylinder  of  the  directrix  of  the  convolute, 
they  will  cut  the  surface  in  involutes  of  circles  equal  to 
those  cut  from  the  surface  by  the  plane  of  the  base.  The 
intersections  of  these  involutes  with  the  respective  lines 
cut  from  the  given  plane,  will  be  points  on  the  curve  of 
intersection  required.  And  this  method  of  solution  may 
be  employed,  when  it  is  found  convenient  to  make  a  tem- 
plate to  easily  transfer  the  different  involutes.  The  origin 
of  the  involute,  or  point  of  contact  with  the  cylinder  of  the 
convolute,  for  any  one  auxiliary  plane,  will  be  where  the 
trace  of  the  auxiliary  plane  cuts  the  helical  curve  upon  the 
cylinder. 

133.    To  pass  a  plane  tangent  to  a  helical  convolute  and 
to  contain  a  point  outside. 

Analysis:— T^e  tangent  plane  will  contain  a  line 
through  the  given  point,  ivhich  will  he  tangent  to  a  curve  of 
section  of  the  surface  hy  a  plane  containing  the  point,  and 
it  will  also  contain  an  element  of  the  surface  passing 
throngh  the  point  of  tangency  of  this  line.  Hence,  pass  a 
secant  plane  through  the  point  to  cut  the  surface  in  a  curve; 
draw  a  tangent  to  this  curve,  through  the  point.  Also  draw 
an  element  of  the  surface  passing  through  the  point  of 
tangency,  the  plane  of  these  two  lines  is  the  plane  required. 

Construction:— Let  the  helical  convolute  be  given  as  in 
Figure  65,  with  the  point  P  as  shown. 


134 


DESCRIPTIVE    GEOMETRY 


FIGUBB    65, 


For  convenience  pass  a  plane  T  through  P  parallel  to 
H.  Its  V  trace  is  parallel  to  the  G.L.  and  the  point  A  in 
which  it  cuts  the  helical  directrix  is  the  origin  of  the 
involute  of  a  circle  in  which  the  surface  cuts  this  plane. 
Through  the  point  p  draw  a  tangent  to  this  involute  as 
plj;  its  V  projection  is  parallel  to  the  G.L.  and  coincides 
with  TV;  through  the  point  of  tangency  B,  draw  an 
element  of  the  surface,  BC;  C  is  its  H  trace.  The  H  trace 
of  the  required  plane  R  goes  through  C  and  is  parallel  to 


SINGLE  CURVED  SURFACES  135 

the  tangent  ph.  The  V  trace  goes  through  the  V  trace  of 
the  element  BC,  and  also  that  of  the  tangent  PB.  As  two 
tangents  can  be  drawn  from  the  point  P  to  the  involute  so 
there  can  be  two  tangent  planes  to  the  convolute  surface 
through  the  point  P.  S  will  be  the  other  plane  as  shown, 
containing  the  tangent  PQ. 

134.    To  draw  a  plane  tangent  to  a  helical  convolute  and 
parallel  to  a  given  line  outside. 

Analysis:— T^e  required  plane  contains  a  line  parallel 
to  the  given  line.  Now,  a  plane  passed  through  the  given 
line  and  conceived  to  he  parallel  to  the  required  plane  may 
also  he  thought  of  as  tangent  to  a  right  cone  with  hase  per- 
pendicular to  the  axis  of  the  convolute,  and  whose  elements 
make  the  same  angle  with  the  plane  of  the  hase,  that  the 
elements  of  the  convolute  do  with  a  plane  perpendicular  to 
its  axis.  Therefore,  choose  any  point  of  the  given  line  to  he 
the  apex  of  a  right  cone  whose  hase  shall  lie  in  a  plane 
which  is  a  right  section  of  the  helical  convolute.  Pass  a 
plane  through  this  line  and  tangent  to  the  cone,  fhy  Section 
118).  The  required  tangent  plane  will  have  its  traces 
parallel  to  this  one  and  may  he  located  hy  aid  of  the  element 
of  tangency  as  in  the  preceding  prohlem. 

Construction:— Let  the  convolute  be  given  as  in  Fig.  QQ 
and  the  line  as  AB.  It  is  seen  from  the  element  of  the 
convolute,  OP,  which  is  parallel  to  V,  that  the  angle  of  the 
elements  with  the  H  plane  is  a,  hence,  through  any  point  of 
AB,  as  B,  draw  a  right  circular  cone  whose  base  is  in  H 
and  whose  elements  make  an  angle  of  a  with  H.  Then, 
through  AB  pass  a  plane  T  tangent  to  the  cone.  Its  H 
trace  goes  through  C,  the  H  trace  of  AB,  and  its  V  trace 


136 


DESCRIPTIVE     GEOMETRY 


SINGLE  CUBVED  SURFACES  137 

contains  the  Y  trace  of  the  line  AB.  Another  plane  R  can 
be  drawn  tangent  to  the  cone  to  contain  the  other  tangent 
to  the  base  from  C. 

Since  the  base  of  the  auxiliary  cone  has  been  taken  in 
H,  the  H  traces  of  the  possible  parallel  tangent  planes  to 
the  convolute  will  be  tangents  to  the  curves  of  intersection 
of  the  surface  with  H  as  shown.  There  can  be  two  planes 
K  and  L  parallel  to  T  and  two  planes  M  and  N  parallel  to 
R.    The  V  traces  are  obtained  by  one  of  the  usual  methods. 

135.    To  develop  the  surface  of  a  helical  convolute. 

Analysis: — Assume,  for  simplicity  in  demonstration, 
that  the  convolute  is  one  whose  axis  is  parallel  to  V  and 
perpendicular  to  H  as  discussed  in  Section  130,  and  let  the 
curve  of  intersection  of  the  surface  with  H,  he  called  the 
base. 

Since  the  helical  directrix  makes  a  constant  angle  with 
the  plane  of  the  base,  (Section  130),  and  since  the  elements 
retain  the  same  length  in  the  development,  and  also  the 
angles  between  consecutive  elements  are  equal,  the  length 
of  the  base  remains  constant,  and  the  helical  directrix 
develops  into  a  curve  of  uniform  curvature.  This  curve 
will  be  a  circle,  and  the  developed  base  will  he  the  involute 
of  this  circle. 

The  mathematical  relations  for  determining  the  radius 
of  curvature  of  the  developed  helical  directrix  may  be 
stated  as  follows : 

E  =  -^ 

cos-  a 

when  a  =  radius  of  the  cylinder  on  which  the  helix  is 
wound  and  a  the  angle  a  tangent  to  the  helix,  makes  with 
a  plane  of  right  section. 


138  DESCRIPTIVE     GEOMETRY 

But  the  radius  may  be  obtained  graphically  as  follows : 
At  the  point  p '  of  Fig.  66,  draw  a  horizontal  line,  at  the  in- 
tersection of  the  tangent  o'p'  with  the  contour  of  the  cylin- 
der draw  a  normal  to  the  tangent  to  intersect  the  horizontal 
line  through  p ' .  The  distance  from  the  intersection  to  p ' 
is  the  radius  of  the  circle  representing  the  developed 
helical  directrix.  The  developed  surface  will  be  limited  by 
this  circle  and  its  involute. 


CHAPTER   VI. 


WARPED    SURFACES. 

136.    Warped  Surfaces. 

Warped  surfaces  have  already  been  defined  as  those 
generated  by  a  line  moving  so  that  no  two  of  its  successive 
positions  intersect  each  other,  which  is  equivalent  to 
saying  that  they  do  not  lie  in  the  same  plane. 

Any  two  elements  of  one  system  of  generation  are 
what  is  known  as  windschief.  The  motion  of  a  line,  in 
generating  these  surfaces,  while  infinite  in  variety  is,  in 
general,  however,  specified  as  being  controlled  by  lines, 
curved  or  straight,  or  by  surfaces  or  both.  That  is,  it 
moves  so  as  to  touch  a  line  in  a  certain  way,  or  a  surface, 
or  be  parallel  to  the  respective  elements  of  a  surface.  One 
of  the  laws  in  illustration  is  that  of  a  line  which  moves  so 
as  to  touch  three  right  line  directrices  which  themselves 
do  not  intersect.  Another  is  one  in  which  a  line  moves 
so  as  to  touch  a  given  right  line  and  a  given  curve  and 
at  the  same  time  be  parallel  to  a  given  plane,  or  still 
another  to  make  a  fixed  angle  with  the  given  right  line. 

To  summarize  the  limiting  conditions,  a  line  may  move 
(1)  so  as  to  have  its  successive  positions  parallel  to  a 
plane;  (2)  so  that  its  successive  positions  are  parallel  to 
the  elements  of  a  cone*;  (3)  so  that  its  successive  posi- 
tions intersect  a  given  line. 

*The  cone  Is  not  limited  to  a  right  circular  cone. 


140 


DESCRIPTIVE     GEOMETRY 


Under  (1)  we  have  (a)  two  right  line  directrices,  (&) 
two  curved  directrices,  (c)  one  straight  line  directrix  and 
one  curved  line  directrix.  Under  (2)  we  have  (a)  two 
straight  line  directrices,  (h)  two  curved  directrices,  (c) 
one  straight  line  directrix  and  one  curved  line  directrix. 
Under  (3)  we  have  a  unique  class  in  which  there  are  two 
curved  directrices  and  one  straight  line  directrix. 

The  classification  may  be  tabulated  as  follows,  with  ex- 
amples of  the  different  kinds  of  surfaces : 

137.    Table  of  classification  of  warped  surfaces. 


Warped 
Surfaces 


Directrix      Directrices 


Examples 


Plane 


Cone 


Line 


2  straight  lines— Hyperbolic 

paraboloid 
2  curved  lines  — Eight  helicoid 
1  straight  line 
1  curved  line 


Conoid 


2  straight  lines— Hyperboloid  of 

revolution* 
2  curved  lines  —  Oblique  heli- 
coid 
1  straight  line  )  Elliptical 
L 1  curved  line    )      hyperboloid 


2  curved  lines  \  Obhque  helicoid 
( 01  varymg  pitch 


♦The  hyperboloid  of  revolution  will,  for  convenience,  be  discussed  under 
surfaces  of  revolution. 


WARPED    SURFACES  141 

138.    The  Hyperbolic  Paraboloid. 

If  a  lines  moves  so  that  it  touches  two  windschief 
right  lines  and  remains  at  the  same  time  parallel  to  a 
given  plane,  the  surface  generated  is  know  as  the  hyper- 
bolic paraboloid,  so  designated  because  sections  of  the  sur- 
surface  are  either  right  lines,  hyperbolas  or  parabolas. 

If  two  lines,  not  in  the  same  plane,  are  intersected  by  a 
series  of  parallel  planes,  the  lines  are  divided  into  propor- 
tional parts  by  the  planes,  from  geometry,  and  if  the  cor- 
responding points  of  division  on  each  are  connected  they 
constitute  the  elements  of  a  hyperbolic  paraboloid. 

Conversely,  if  any  two  lines  not  in  the  same  plane  are 
divided  into  proportional  parts,  the  light  lines  joining  the 
corresponding  points  of  division  on  each  will  lie  in  parallel 
planes  and  be  the  elements  of  a  hyperbolic  paraboloid,  the 
plane  directer  of  which  is  parallel  to  any  two  of  them. 

Thus  let  AB  and  CD,  Fig.  67,  be  any  two  non-intersect- 
ing lines.  With  any  radius  as  a-1  divide  ab  into  any  num- 
ber of  equal  parts ;  with  any  other  radius  as  d-6  divide  d-c 
into  the  same  number  of  equal  parts,  find  the  vertical  pro- 
jections of  these  points  of  division.  Join  the  correspond- 
ing points  of  division  on  these  lines  as  1-d^  2-12,  3-11,  etc. 
They  are  the  elements  of  the[surface,  and  it  is  seen  that  the 
curve  which  is  the  envelope  of  its  tangents  is  the  parabola. 
For  from  geometry,  any  element  as  9-5,  being  a  tangent  to 
the  parabola,' divides  the  two  other  tangents  AB  and  CD 
proportionally,  a  well  known  property  of  the  parabola. 
The  lines  AB  and  CD  are  the  two  linear  directrices  of  the 
surface  and  are  shown  here  as  oblique  to  the  coordinate 
planes. 

The  characteristic  properties  of  this  surface  are  best 


142 


DESCRIPTIVE    GEOMETRY 


Figure  67. 


brought  out  by  projecting  the  elements  of  the  surface  upon 
two  coordinate  planes  which  are  not  necessarly  at  right 
angles  to  each  other,  one  parallel  to  the  two  rectilinear 
directrices  which  we  will  call  Q,  the  other  parallel  to  the 
elements  of  the  one  system  of  generation,  or  in  other  words 
the  plane  directer  of  the  system  which  we  will  call  P.  Let 
Fig.  68  represent  these  planes  seen  edgewise,  in  other 
words  projected  upon  an  end  vertical  plane.  The  project- 
ing lines  of  any  point  M  will  be  parallel  respectively  to  the 


WARPED    SURFACES  143 


FiGUBB    68. 

coordinate  planes  —  although  not  perpendicular  to  each 
other,— and  parallel  also  to  the  end  plane,  so  that  when  the 
planes  P  and  Q  are  unfolded,  the  projecting  lines  will  con- 
stitute one  and  the  same  perpendicular  to  the  G.  L. 

139.  *  THEOREM  XIL 

The  projections  of  all  the  elements  of  one  system  of 
generation  of  a  hyperbolic  paraboloid,  upon  the  plane 
directer  intersect  each  other  in  a  point,  known  as  a  '  point 
of  concourse.* 

Proofi — Let  Figure  69  represent  the  projections  of  the 
two  directrices  AB  and  CD,  upon  two  inclined  coordinate 
planes,  Q  and  P,  respectively,  and  let  P  be  taken  as  a 
horizontal  plane.  Since  the  plane  Q  is  parallel  to  both 
of  the  directrices,  their  P  projections  will  be  parallel  to 
the  G.L.,  and  they  may,  with  this  proviso,  be  assumed 
anywhere  as  shown.  Let  the  point  of  intersection  of 
the  Q  projections  of  AB  and  CD,  be  denoted  by  y"^  — 
merely  an  arbitrary  point  and  not  standing  for  the  pro- 
jection of  any  point  in  space. 

Assume  any  plane  parallel  to  P,  its  vertical  trace  being 
TQ.    It  will  pierce  the  directrices  in  two  points,  respec- 

*  After  Watson's  'Descriptive  Geometry',  page  41. 


144  DESCRIPTIVE     GEOMETRY 

lively,  E  and  F,  which,  if  connected  by  a  line,  will 
constitute  an  element  of  the  surface.  Drop  a  perpendicu- 
lar to  the  G.L.  through  y"^,  and  let  its  intersection 
with  Pe^  be  called  x^  also  its  intersection  with  the 
G.L.,  be  called  1^  and  with  the  P  projection  of  the 
directrix  AB  as  ^,  and  with  TQ  as  5.  And  let  the  inter- 
section of  a  perpendicular  to  the  G.L.  through  f^  with 
the  P  projection  of  one  of  the  directrices  be  called  4, 
From  the  similar  triangles  e''x''2,  eV4,  h'^y^'l, 
e^y^S,    y''ld\  and  y^'Sf,    we  have— 

P4         €"4        eV^        b'^d'^ 


h 


but  the  ratio  ^  q  ^q     is  independent  of  the   position  of 
the  generatrix  FE,  that  is,  the  cutting  plane  T,  hence 

x'2  _  yi 

7^ Wd^  ^  constant 

and  a;^^  is  a  constant  for  all  positions  of  the  generatrix- 
By  similar  reasoning,  it  also  may  be  proved  that  the 
projections  of  the  other  system  of  generation,  upon  the 
plane  directer  Q,  pass  through  a  point  on  that  plane,  which 
point  is  the  intersection  of  the  Q  projections  of  the 
directrices  AB  and  CD,  denoted  y'^ . 

140.  THEOREM  XIIL 

The   section  of    a    hyperbolic  paraboloid   by    a    plane 
parallel  to  tiie  two  rectilinear  directrices  is  a  straight   line. 

Proof:— Let  the  surface  be  represented  as  in  Fig.  69, 
and  assume  R  to  be  a  plane  parallel  to  Q,  the  plane 
which  in  turn  is    parallel  to  the  two  directrices.    The 


WAKPED    SURFACES 


145 


Figure    69. 


generatrix  FE  meets  this  plane  R  in  the  point  G.    From 
similar  triangles  we  have: 


e'g' 


?^ 


But  we  see  that  this  ratio  would  be  true  for  any  position 

of  the  elements  EF,  that  is  the  ratio     q    »   is  a  constant 

and  the  locus  of  the  various  intersections  of  elements 
of  the  surface  with  any  plane  R  would  be  a  straight  line 
which  goes  through  the  point  y  ** . 

Corollary  1:— For  every  hyperbolic  paraboloid  there  are 
two  systems  of  generation.  For  since  any  plane  parallel  to 
Q  will  cut  the  surface  in  a  line  whose  P  projection  is  par- 
allel to  the  G.L.,  the  plane  Q  will  be  the  plane  directer  of 


146  DESCRIPTIVE     GEOMETRY 

the  system  of  such  intersections  or  elements  and  the  pro- 
jections of  these  elements  upon  the  plane  Q  pass  through 
a  common  point  y "" ,  just  as  those  belonging  to  the  other 
system  pass  through  x^  with  P  as  the  plane  directer. 

Corollary  2:— T/^e  generatrix  of  one  system  in  a  Jiyper- 
holic  paraboloid  intersects  every  element  of  the  other  system. 
For  the  projection  of  any  two  generatrices,  one  of  each 
system,  upon  either  coordinate  plane  P  or  Q  will  intersect 
each  other  in  a  point  which  can  be  the  projection  of  but 
one  point  upon  the  surface.  Hence,  any  two  elements  of 
one  system  may  be  taken  as  the  directrices  for  the  other 
system.  It  follows  also  that  through  any  point  of  the 
surface  two  lines  may  be  drawn  which  are,  respectively, 
the  elements  of  each  system  of  generation. 

Corollary  3:— No  two  generatrices  of  the  same  system  in 
a  hyperbolic  paraboloid  intersect  each  other  for  we  have 
seen  that  the  projection  of  one  system  upon  one  of  the  co- 
ordinate planes  P  and  Q  are  lines  parallel  to  the  G.  L.  and 
in  the  corresponding  projection  they  are  oblique  and  make 
different  angles  with  the  G.  L. 

Corollary  4: — The  hyperbolic  paraboloid  may  be  defined 
as  the  surface  generated  by  a  line  which  moves  so  as  to  touch 
three  other  non-intersecting  lines  all  of  which  are  parallel  to 
a  common  plane  and  divide  said  lines  into  proportional 
parts  for  since  an  element  of  one  system  intersects  every 
element  of  the  other  system  the  surface  may  have  three 
rectilinear  directrices  instead  of  two  rectilinear  directrices 
and  a  plane  directer. 

It  was  noted  in  Theorem  XIII  that  for  the  first  system 


WARPED    SURFACES  147 

of  generation  discussed  there  was  a  point  in  the  projection 
of  the  elements  upon  the  plane  directer  which  is  common 
to  the  projection  of  all  the  elements  on  that  plane,  if  ,  and 
for  the  second  system,  similarly,  a  point  in  the  projection 
of  these  elements  upon  their  plane  directer,  which  is 
common  to  all  these  elements,  a;p.  These  points  are  known 
as  ^points  of  concourse\  It  may,  moreover,  be  seen  that 
each  point  of  concourse,  as  a  point  in  one  system,  is  the 
projection  upon  that  plane  directer  of  an  element  of  the 
surface  belonging  to  the  other  system  of  generation  which 
coincides  with  the  projecting  line  of  that  element  upon  the 
plane.  For  example,  referring  to  Figure  68,  the  line 
Mw^  is  the  projection  of  such  an  element  parallel  to  Q, 
upon  the  plane  P,  and  Mm"  is  the  projection  of  an  element 
parallel  to  P  upon  the  plane  Q. 

The  trace  of  the  hyperbolic  paraboloid  upon  the  plane 
directers  P  and  Q  is  found  as  follows :  —  The  P  trace  goes 
through  the  P  traces  of  the  directrices  AB  and  CD,  see 
Fig.  70,  and  since  an  element  of  the  surface  is  projected 
upon  Q  in  the  point  t/"  that  being  also  its  Q  trace  the  Q 
trace  of  the  plane  of  the  hyperbolic  paraboloid  goes 
through  that  point  SQ  and  SP  will  be  its  respective  traces. 
If  the  traces  did  not  meet  in  the  G.  L.  within  the  limits  of 
the  drawing,  then  the  Q  trace  of  any  auxiliary  element  of 
the  surface  could  be  found  and  the  required  traces  located. 

141.    To  assume  a  point  upon  the  surface  of  a  hyperbolic 
paraboloid. 

Analysis: — Assuming  either  projection  of  the  point, 
draw  through  it  any  element  of  one  system  of  generation. 
The  corresponding  projection  of  the  point  will  he  found  upon 


148 


DESCRIPTIVE  GEOMETRY 


the  corresponding  projection  of  the  element,  and  the  cor- 
responding projection  of  the  element  can  he  found  by  its 
intercepts  upon  the  directrices. 


FiGUEE     70. 


Construction:  —  Let  the  surface  be  represented,  as  in 
Fig.  70  and  let  i^  be  the  projection  of  an  assumed  point, 
draw  through  ^'^  and  the  point  of  concourse,  x^ ,  an  element 
of  the  system  of  generation  which  is  parallel  to  P.  Find 
by  its  intersection  with  the  P  projection  of  the  directrices, 
the  Q  projection  of  this  element.  ^'*  is  found  at  the  inter- 
section of  a  perpendicular  to  the  G.  L  through  i^  and  the 
Q  projection  of  the  element. 

If  i"^  had  been  assumed,  draw  the  Q  projection  of  an 
element  through  the  point  belonging  to  the  system  which 
is  parallel  to  P  by  making  its  Q  projection  parallel  to  the 
G.L.    The  intercepts  with  the  directrices  will  give  the  P 


WARPED    SURFACES  149 

projection  and  from  that  the  P  projection  of  the  point  may 
be  found  upon  the  P  projection  of  the  element  through 
the  point. 

Note: — An  element  drawn  through  Vy"^  will,  in  P  pro- 
jection, pass  through  i^ ,  and  be  parallel  to  the  G.L.,  and 
this  element  may  equally  well  be  used  to  obtain  the  cor- 
responding projection  of  the  assumed  point. 

Tangent  planes  to  warped  surfaces  have  the  same 
peculiarities  as  those  to  other  surfaces  which  have  already 
been  discussed,  that  is,  they  are  the  locii  of  lines  which 
can  be  drawn  tangent  to  curves  cut  frpm  the  surface.  An 
element  of  a  surface  which  has  straight  line  elements  is  a 
convenient  line  of  the  plane  to  obtain.  In  warped  surfaces 
a  tangent  plane  while  it  may  contain  an  element  or 
elements  of  the  surface  will  only  be  tangent  in  general,  at 
a  point  of  the  surface,  for  the  successive  elements  are  non- 
intersecting,  and  a  plane  passed  through  one  will  inter- 
sect the  next  consecutive  one  in  a  point.  Therefore  the 
tangent  plane  at  a  given  point  is  derived  conveniently  by 
an  element  through  the  point  and  a  tangent  to  the  curved 
section  through  the  point.  If  the  surface  is  doubly  ruled 
like  the  hyperbolic  paraboloid,  the  tangent  plane  will  con- 
tain the  two  elements  of  the  surface,  one  of  each  system 
of  generation,  intersecting  in  the  point. 

142.    To  pass  a  plane  tangent  to  a  hyperbolic  paraboloid 
at  a  point  on  the  surface. 

Analysis:— T^e  tangent  plane  must  contain  an  element 
of  the  surface;  but  through  every  point  of  the  surface,  there 
pass  two  elements  one  belonging  to  each  system  of  genera- 
tion,  hence  the  tangent  plane  will  contain  both  and  be 


150  DESCRIPTIVE    GEOMETRY 

tangent  to  the  surface^  only  at  a  point.  Therefore,  through 
the  given  point,  draw  an  element  of  each  system  of  genera- 
tion. The  plane  of  the  elements  is  the  plane  required. 
The  curves  cut  from  the  surface  hy  the  tangent  plane,  are 
the  two  elements  lying  in  the  tangent  plane. 

Let  any  generatrix  of  the  system  P,  that  is  the  one  of 
which  P  is  the  plane  directer,  revolve  about  the  point  of 
concourse,  x^ ,  occupying  different  possible  positions  in  the 
surface,  see  Fig.  71.  It  will  assume  positions  x^  i^ ,  f^  e^ , 
d^  If  ,  etc.,  respectively.  When  it  occupies  a  position 
parallel  to  the  G.L.,  its  Q  projection  would  be  at  an 
infinite  distance  below  the  Gr.L.  If  it  continues  to  revolve, 
it  would  reappear  again  from  an  infinite  distance  above 
the  G.L.  on  the  plane  Q.  When  its  P  projection  is  per- 
pendicular to  the  G.L.,  its  Q  projection  is  the  point  y"^ , 

Therefore  if  through  any  point  in  the  plane  directer 
of  one  of  the  systems  we  draw  lines  parallel,  respectively, 
to  the  elements  of  that  system,  they  will  occupy  all  possi- 
ble positions  around  that  point,  and  the  one  infinitely 
distant  will  be  parallel  to  the  G.L.  And  since  there  is 
in  each  system  an  element  infinitely  removed  and  parallel 
to  the  Gr.L.,  it  is  evident  that  an  element  can  be  found  to 
make  any  assigned  angle  with  the  line  of  intersection  of 
the  plane  directers. 

The  two  elements,  one  of  each  system,  whose  projections 
are  the  points  of  concourse,  respectively,  intersect  each 
other  in  a  point  of  the  surface  known  as  the  vertex. 

The  two  diametral  planes  of  the  surface  are  known  as 
those  respectively  which  (1),  are  perpendicular  to  each 
other  and  go  through  the  vertex  perpendicular  to  the  G.L., 


WARPED    SURFACES  151 

or  intersection  of  the  plane  directers,  and  (2),  go  through 
the  vertex  and  bisect  the  dihedral  angle  between  the  two 
plane  directers. 

If  the  surface  is  projected  upon  the  coordinate  planes 
in  such  a  manner  that  H  is  parallel  to  the  first  mentioned 
diametral  plane  and  Y  is  parallel  to  the  second  mentioned 
diametral  plane,  we  will  obtain  projections  which  will 
facilitate  the  drawing  of  elements  of  the  surface,  tangent 
planes,  sections,  etc. 

143.  (a)  To  draw  the  projections  of  the  hyperbolic 
paraboloid  upon  planes  parallel,  respectively,  to  the  dia- 
metral planes,  (b)  to  find  the  sections  of  the  surface  made 
by  planes  parallel  to  the  coordinate  planes,  (c)  to  construct 
a  tangent  plane  at  any  given  point 

Analysis:— T/^e  analysis  of  the  first  part  of  the  problem 
is  analogous  to  the  explanation  given  in  Section  139  and 
need  not  he  revealed. 

Any  secant  plane  will  cut  either  straight  lines,  hyper- 
bolas, or  parabolas  from  the  surface.  Secant  planes  parallel 
to  H,  namely,  perpendicular  to  the  plane  directers,  will  cut 
hyperbolas  of  equal  nappes,  and  one  will  cut  two  straight 
lines.  Any  secant  plane  parallel  to  V,  namely,  parallel  to 
the  plane  bisecting  the  dihedral  angle  between  the  plane 
directers  will  cut  the  surface  in  a  parabola. 

Any  tangent  plane  through  a  given  point  is  obtained  by 
getting  the  traces  of  the  plane  of  the  two  elements  one  of 
each  system  intersecting  in  the  point 

Construction:— See  Figure  71.  The  H  plane  is  taken 
perpendicular  to  the  two  plane  directers  P  and  Q,  and  the 
V  plane  is  taken   parallel  to  the  plane  bisector  of  the 


152 


DESCRIPTIVE    GEOMETRY 


WARPED    SURFACES  153 

dihedral  angle  between  P  and  Q.  It  can  be  seen  that  the 
projections  of  both  systems  of  elements  upon  the  H  plane 
will  be  parallel  to  each  other,  respectively.  AB  and  CD 
are  the  two  directrices.  These  are  divided  proportionally 
into  twelve  equal  parts  and  the  corresponding  points 
of  division  joined,  giving  as  elements  of  the  one  system 
1-11,  2-10,  etc.  By  joining  AC  and  BD,  and  similarly 
dividing  these  into  twelve  equal  parts  we  get  elements 
1-11,  2-10,  etc.,  of  the  other  system  of  generation.  These 
are  shown  also  projected  upon  an  end  plane,  for  conveni- 
ence revolved  into  H.  The  directrices  AB  and  CD  in  V 
projection  are  for  convenience  assumed  as  equally  inclined 
to  the  G.L. 

A  parabola  is,  in  the  V  projection  as  in  Figure  71,  the 
envelope  of  the  tangents,  and  is  the  curve  of  intersection 
with  the  surface  of  a  plane  parallel  to  V,  *  its  points  cor- 
responding to  those  in  which  the  elements,  in  H  pro- 
jection, intersect  the  line  he.  For,  referring  to  the  V 
projection  and  the  tangent  7-5,  together  with  AB  and  CD. 

c'5   _  £2 
W        7h' 

Hence,  the  tangent  7-5  divides  the  tangents  AB  and 
CD  into  proportional  parts,  a  well  known  property  of  the 
parabola. 

Assume  a  horizontal  plane  to  cut  the  surface,  shown 
by  its  trace  TV.  Noting  the  points  in  which  the  respective 
elements  pierce  this  plane,  and  getting  their  respective  H 
projections,  it  is  found  that  they  lie  on  a  hyperbola  of  two 
nappes,  symmetrical  with  respect  to  the  line  connecting  a 
and  d.  Assume  another  V  plane,  shown  by  its  trace  UY, 
and  similarly  plotting  the  traces  of  elements  with  it,  we 
find  that  the  curve  of  intersection  is  another  hyperbola. 


154  DESCRIPTIVE    GEOMETRY 

with  two  nappes  symmetrical  with  the  hne  connecting  h 
and  c.  The  asymptotes  to  the  hyperbola  can  be  found  by 
taking  a  horizontal  section  of  the  surface,  and  passing 
through  the  vertex  of  the  parabola  in  Y  projection. 

A  section  of  the  surface,  by  a  plane  oblique  to  H, 
would  give  hyperbolas  of  unequal  nappes. 

Let  it  be  required  to  draw  a  tangent  plane,  at  the 
point  M,  the  intersection  of  the  elements  7-5  and  8-4,  one 
belonging  to  one  system  of  generation,  and  the  other 
belonging  to  the  other  system.  The  plane  of  these  two 
elements  is  the  plane  required,  shown  by  its  traces  EV  and 
EH.  It  can  be  seen,  that  it  is  also  a  secant  plane  of  the 
surface,  being  tangent  only  at  the  point  M. 

The  vertex  of  the  surface  is  the  point  of  intersection  of 
two  elements,  one  of  each  system,  whose  plane  is  perpen- 
dicular to  both  plane  directers,  i.e.,  perpendicular  to  their 
line  of  intersection.  Such  a  point  is  the  intersection  of 
elements  6-6  of  each  system  in  Figure  71. 

The  axis  of  the  surface  is  a  line  through  the  vertex, 
parallel  to  both  plane  directers,  i.e.,  parallel  to  their  line 
of  intersection. 

If  the  angle  between  the  plane  directers  is  a  right 
angle,  the  surface  is  known  as  a  right  or  rectangular 
hyperbolic  paraboloid.  If  the  angle  is  not  a  right  angle, 
the  surface  is  known  as  an  oblique  hyperbolic  paraboloid, 

144.    To  assume  a  point  upon  the  surface  of  the  hyper- 
bolic paraboloid  as  in  Section  141. 

Analysis:— TAe  horizontal  projection  of  the  point  may 
be  assumed,  and  an  element  of  the  surface  drawn  through 
it  from  whence,  the  V  projection  may  be  obtained.    The  V 


WARPED    SURFACES  155 

projection  of  the  point  may  he  assumed;  in  this  case  pass  a 
horizontal  secant  plane  through  it,  to  cut  a  hyperbola  from 
the  surface,  whence  the  H  projection  of  the  point  will  he 
found  upon  the  hyperbola.  The  assumed  V  projection 
stands  for  two  points  upon  the  surface,  equally  distant  in 
front  of  and  behind  the  vertex. 

145.  To  pass  a  plane  through  a  given  right  line  and  tan- 
gent to  a  hyperbolic  paraboloid. 

Analysis:— T/^e  tangent  plane  contains  an  element  of 
the  surface,  in  addition  to  the  given  line.  Therefore, 
through  the  piercing  point  of  the  given  line,  and  the  surface, 
draw  an  element  of  the  surface;  the  plane  of  these  tivo  lines 
is  the  plane  required.  It  will  be  tangent  to  the  surface  at 
some  point  of  the  element. 

The  pilot  or  cow-catcher  of  some  locomotives  is  built 
upon  the  lines  of  a  hyperbolic  paraboloid.  The  prows  of 
some  boats  illustrate  it  also. 

146.  The  Conoid  is  a  warped  surface  having  a  straight 
line,  and  a  curved  line  directrix,  and  a  plane  directer. 

A  right  conoid  is  a  conoid  in  which  the  plane  directer 
is  perpendicular  to  the  straight  line  directrix  and  the  plane 
of  the  curvilinear  directrix.  It  may  be  exemplified  by  a 
line  which  moves  so  as  to  touch  a  straight  line  and  a  given 
circle,  the  given  line  being  parallel  to  the  plane  of  the 
circle,  and  the  plane  directer,  perpendicular  ,to  both, 
whence  we  have  a  surface  as  represented  in  Fig.  72.  The 
plane  directer  is  an  end  vertical  plane. 

A  right  helical  conoid  is  a  conoid  in  which  the  curvi- 
linear directrix  is  a  helix,  and  the  straight  line  directrix 


156  "  DESCRIPTIVE     GEOMETRY 

the  axis  of  the  helix.    It  is  exemphfied  in  the  -face  of  a 
square  threaded  screw. 

An  oblique  helical  conoid  is  a  warped  surface  in  which 
the  generatrix,  touching  the  helix  and  its  axis,  is  oblique 
to  the  latter.    It  is  exemplified  in  the  V  threaded  screw. 

147.  To  draw  a  conoid  having  a  circular  directrix,  and  a 
straight  line  directrix  parallel  to  the  plane  of  the  circle  and 
a  plane  directer  perpendicular  to  both;  to  assume  a  point  on 
it,  and  to  draw  a  tangent  through  the  point 

Analysis: — Assume  one  coordinate  plane  parallel  to  the 
plane  directer,  and  one  perpendicular  to  it,  and  also  one 
parallel  to  the  circular  directrix.  The  projection  of  the 
elements  upon  the  second  plane  will  he  parallel  to  the  G,L,, 
and  their  projections  upon  the  first  will  go  through  a  com- 
mon point  which  is  the  projection,  upon  that  plane,  of  the 
straight  line  directrix. 

To  assume  a  point  on  the  surface,  assume  either  pro- 
jection and  draw  an  element  through  the  point. 

To  pass  a  plane  tangent  to  the  surface  at  any  point, 
draw  an  element  through  the  point,  and  it  will  be  contain- 
ed within  the  tangent  plane.  The  tangent  plane  will  also 
contain  a  tangent,  at  the  point  of  contact,  of  a  curve  of 
section  of  the  surface  made  by  any  secant  plane.  The 
plane  of  these  two  lines  is  the  plane  required. 

Construction:— The  conoid  is  shown  in  projection  in 
Fig.  72.  The  circular  directrix  is  placed  in  H  and  the 
straight  line  directrix  perpendicular  to  Y,  which  is  the 
plane  directrix.  For  convenience,  the  elements  are  taken 
as  touching  the  circular  directrix  at  equally  distant  points, 
1,  2,  3,  4,  etc.    o'p'  is  the  vertical  projection  of  the  direct- 


WARPED   SURFACES 


wr 


Figure   72. 


rix,  seen  as  a  point.  This  directrix  is  also  an  axis  of  the 
surface.  Another  axis,  more  generally  considered  as  the 
axis,  is  a  line  through  the  center  and  perpendicular  to  the 
plane  of  the  circular  directrix. 

Planes  perpendicular  to  this  axis  cut  the  surface  in 
ellipses. 

To  assume  a  point  on  the  surfaee:— Assume  either 
projection,  say  the  H  projection  as  m.  Draw  an  element 
through  the  point,  and  by  its  vertical  projection,  obtain  the 
y  projection  of  the  point.  The  assumed  projection  of  the 
point,  in  any  case,  would  stand  for  two  points  on  the  sur- 
face; if  the  H  projection,  then  a  point  on  either  nappe-  if 


158  DESCRIPTIVE     GEOMETRY 

the  V  projection  then  a  point  upon  either  the  front  or  back 
of  the  surface. 

To  draw  a  tangent  plane  to  the  surface,  say  at  the 
point  M.  Draw  an  element  of  the  surface  through  the 
point;  it  will  be  contained  within  the  tangent  plane,  Pass 
a  plane  through  the  point,  to  cut  a  curve  from  the  surface, 
preferably  a  plane  parallel  to  H  as  T.  The  tangent  to  the 
ellipse  of  section  at  M  will  also  be  a  line  of  the  plane.  The 
y  trace  of  the  required  plane,  R,  goes  through  the  Y  trace 
of  this  line,  and  is  parallel  to  the  element  of  the  surface, 
through  the  point  of  tangency.  The  H  trace  is  parallel  to 
the  tangent  to  the  curve  of  section. 

This  plane  will  not  be  tangent  to  the  surface  all  along 
an  element,  because  sections  of  the  surface,  at  various 
points  to  intersect  an  element,  through  any  given  point, 
will  not  have  tangents  at  those  points  of  intersection 
which  are  parallel. 

The  foregoing  example  is  only  one  of  an  infinite 
number  of  conoids. 

The  curvilinear  directrix  may  he  either  a  curve  of 
single  or  of  double  curvature.  The  straight  line  directrix 
may  occupy  any  position,  with  respect  to  the  curvilinear 
directrix.  For  example,  if,  in  a  conoid  of  the  class  just 
described,  the  straight  line  directrix  had  been  oblique  to 
the  circle,  we  would  have  had  an  oblique  circular  conoid. 

A  right  elliptical  conoid  is  one  in  which  the  conditions 
are  the  same  as  in  the  case  just  discussed,  except  that  the 
curvilinear  directrix  is  an  ellipse. 

148.    The  helicoid  is  a  warped  surface  generated  by  a 
line  which  moves  so  as  to  touch  two  helical  directrices,* 

*A  helix,  as  here  used,  Is  a  term  designating  a  helix  described  upon  a  right 
cylinder. 


WARPED    SURFACES  159 

having  a  common  axis,  and  has  its  successive  positions 
parallel,  either  to  a  plane  directer,  or  to  a  cone  directer. 

A  right  helicoid  is  a  helicoid,  in  which  the  elements 
are  parallel  to  a  plane  directer. 

An  oblique  helicoid  is  a  helicoid  in  which  the  elements 
are  parallel  to  those  of  a  cone,  i.e.,  has  a  cone  directer. 

The  axis  of  the  helical  directrices  is  known  as  the  axis 
of  the  surface. 

A  helicoid  of  uniform  pitch  is  a  helicoid  in  which  both 
helical  directrices  have  the  same  pitch. 

A  helicoid  of  varying  pitch  is  a  helicoid  in  which  the 
helical  directrices  have  different  pitches,  hence,  the 
motions  of  various  points  of  the  genertrix  axially,  vary, 
and  the  angles  of  the  generatrix,  with  the  axis,  vary. 

149.     Figure  73    represents  an  oblique  helicoid.     One 

spire  of  the  helical  directrix,  1,  4,  7,  etc.,  is  shown,  with  its 
axis  parallel  to  Y,  Elements  of  the  surface  may  be  drawn 
as  follows:  Assume  the  angle  of  the  elements,  with  the 
axis  of  the  helical  directrix,  to  be  30'' .  Draw  an  element 
of  the  surface,  which  is  parallel  to  V,  and  will  show  its  true 
angle  with  the  directrix.  Such  an  element  will  be  e' a'. 
Since  the  angle  of  the  elements  with  the  straight  line 
directrix  is  constant,  every  point  of  the  element  will  de- 
scribe a  helix  of  constant  and  equal  pitch;  therefore,  the 
points  of  intersection  of  the  elements,  with  the  straight 
line  directrix,  will  be  distant  from  one  another  the  amount 
of  axial  advance  which  the  points  of  intersection  of  the 
elements,  with  the  helical  directrix,  are  onie  from  another. 
Therefore,  to  draw  any  other  element:  Choose  a  point 
upon  the  helical  directrix,  as  6 '  through  which  the  element 
is  to  pass,  note  its  axial  distance  from  the  point  i,  that  is. 


160 


DESCRIPTIVE    GEOMETRY 


Figure  73 


in  this  case,  the  distance  the  point  is  from  the  H  plane. 
Lay  this  distance  off  on  the  axis,  measuring  upward  from 
the  point  e'  obtaining  6',  then  h' 6'  is  the  V  projection  of 
the  element;  it  intersects  H  in  a  point  C,  a  point  upon  the 
curve  of  intersection  of  the  surface  with  the  H  plane ;  other 
elements  may  be  assumed  in  a  similar  manner. 

The  H  projection  of  an  element  may  be  assumed  as 
readily  as  the  V  projection,  and  from  it  the  Y  projection, 
by  the  method  just  described. 


WARPED    SURFACES  161 

150.  To    assume    a  point    upon    the    oblique  helicoid: 

Assume  one  projection  of  the  point,  as  the  H  projection. 
*Draw  the  projection  of  the  element  through  it  to  intersect 
the  axis  and  the  helical  directrix;  from  the  V  projection  of 
the  latter  point,  note  its  axial  advance  from  any  other 
convenient  point  on  the  directrix,  through  which  an 
element  passes,  and  proceed  as  described  in  the  preceding 
paragraph. 

151.  To   pass  a    plane    tangent    to    an   oblique  helical 
conoid,  at  a  given  point  on  the  surface. 

Analysis: — The  tangent  plane  will  contain  an  element 
passing  through  the  given  point;  it  will  also  contain  a  tan- 
gent to  a  curve  of  section  of  the  surface,  through  the  point, 
which  is  a  helix  of  equal  pitch  hut  lesser  diameter  than  that 
of  the  directrix.  But  the  tangent  to  the  directrix,  at  the 
intersection  of  the  element,  is  in  general  most  convenient. 

Construction:— Let  the  surface  be  given  as  in  Fig.  73, 
and  let  the  point  of  tangency  be  D.  Draw  an  element  of 
the  surface,  through  the  point.  The  H  trace  of  the  tan- 
gent plane  goes  through  the  H  trace  of  this  element.  A 
tangent  DF,  to  the  helical  directrix,  may  next  be  drawn. 
The  H  trace  of  the  tangent  plane  also  goes  through  the  H 
trace  of  this  tangent.  The  V  trace  can  then  be  found,  D 
being  a  point  in  the  plane. 

152.  To  find   the   curve   of  intersection    of  an  oblique 
helical  conoid  by  any  plane. 

Analysis:— T^e   general   conditions  of  surfaces    being 

♦For  convenience  we  will  consider  the  helicoid  as  projected  onH  and  V,  when 
Its  axis  is  perpendicular  to  the  former. 


162 


DESCRIPTIVE     GEOMETRY 


intersected  hy  planes,  can  he  observed  here.  The  auxiliary 
secant  planes,  hy  being  taken  so  as  to  contain  the  axis  of  the 
surface,  will  cut  the  latter  in  straight  lines,  the  elements  of 
the  surface,  and  their  intersections  with  the  given  plane, 
will  give  points  upon  tne  curve  of  intersection. 


Figure   74    and    75. 


153.    The  right  and  oblique  helicoids  are  siiown  in  Figs. 
74  and  75. 

Figure  74  is  the  right  helicoid.  Figure  75  illustrates 
the  oblique  helicoid.  The  moving  element  is  shown  in 
each  figure  as  the  line  AB.  It  is  tangent,  in  each,  to  the 
inner  cylinder,  upon  which  the  helix  is  described,  although 
it  cuts  the  helical  directrix. 


WARPED    SURFACES 


163 


This  surface,  like  many  others,  is  not  adequately  rep- 
resented by  drawing  a  few  positions  of  elements.  It  is 
necessary  to  take  a  limited  element,  and  plot  the  paths 
traced  by  several  points,  on  the  element  and  draw  the  con- 
tour of  which  these  paths  are  the  envelope.  If  this  is 
done,  it  is  seen  that  the  oblique  helicoid  has  the  form  of  an 
augur. 

154.  THEOREM    XIV. 

The  trace  of  an  oblique  helicoid  with  any  plane  peri>en- 
dicular  to  the  axis  of  the  helical  directrix  is  an  archimedian 
spiral 


Figure  76. 

Proof. — Referring  to  Figure  76,  suppose  an  element 
a'h' c'  to  move  half  way  around  the  cylinder.  It  will 
occupy  a  position  e' c'  f\  parallel  toh' d\  since  the  pitch 
is  constant.  Since  h' c'  =  2  o'h\  we  will  have  d'  f  = 
2  o' d'  =  a' d' *  And,  since  the  revolutions  of  points  of 
the  element  are  uniform,  and  the  axial  advance  is  also 
uniform,  the  trace  with  a  plane  perpendicular  to  e'o', 
is  an  equable  spiral,  of  which  d'  f  is  the  radial  expan- 
sion, in  half  a  revolution;  and  for  any  other  position  of 
the  element,  it  is  directly  proportional  to  the  axial 
advance,  and,  since  these  properties  are  identical  with 
the  archimedian  spiral,  the  curve  traced  is  identical. 


164  DESCBIPTIVE     GEOMETRY 

155.  To  ascertain  the  point  of  tangency,  of  any  given 
plane,  with  a  oblique  helicoid,  along  a  given  element 

Analysis — Since  the  tangent  plane  contains  hut  one 
element  of  the  surface,  it  will  pierce  the  other  elements  in 
voints,  which  lie  upon  a  curve,  cutting  the  given  element  in 
the  point  of  tangency. 

156.  The  hyperboloid  of  one  nappe  is  a  unique  warped 
surface  in  that  it  is  also  a  surface  of  revolution,  and  the 
only  one  of  the  latter  class  which  is,  at  the  same  time, 
a  warped  surface.  Since  surfaces  of  revolution  have 
special  characteristics,  controlling  their  treatment  in 
projection,  and  as  the  hyperboloid  has  no  particular  char- 
acteristics, as  a  warped  surface,  beyond  those  which  have 
already  been  discussed,  it  will  be  treated  fully  latter  in 
connection  with  surfaces  of  revolution. 


CHAPTER   VII 


DOUBLE  CUEVED  SUEFACES  AND  SITRFACES 
OF  REVOLUTION. 

157.    Double  curved  surfaces. 

Double  curved  surfaces  are  those  which  can  only  be 
generated  by  the  motion  of  curves ;  they  have  no  straight 
line  elements,  for,  if  they  did,  they  would  belong  either  to 
the  single  curved  surfaces  or  warped  surfaces.  But  the 
converse  is  not  true,  for,  every  surface  generated  by  the 
motion  of  a  curve  is  not  a  double  curved  surface.  A 
double  curved  surface  may  be  fully  defined  as  one  gener- 
ated by  the  motion  of  any  curve,  such  that  no  two 
consecutive  points  of  the  curve  travel  in  straight  line  paths. 

There  is  an  infinite  variety  of  double  curved  surfaces, 
since  there  is  both  an  infinite  variety  of  curves,  plane  and 
double  curved,  and  an  infinite  variety  of  ways  in  which 
the  curves  may  move  to  generate  surfaces.  But  the  most 
familiar  double  curved  surfaces,  and  those  of  interest  to 
the  student,  are  also  surfaces  of  revolution.  No  particular 
interest  attaches  to,  or  value  to  be  gained  by,  the  discus- 
sion of  double  curved  surfaces,  in  general,  except  to  note 
the  following  points : 

Planes  can  he  tangent  to  such  surfaces  only  in  a  point, 
or  isolated  points. 

A    plane  may  he  tangent  to  a  double  curved  surface. 


366  DESCRIPTIVE     GEOMETRY 

and  also  intersect  the  surface  in  a  curve,  which  may  or 
may  not  contain  the  point,  or  points,  of  tangency. 

A  tangent  plane,  if  it  also  cuts  the  surface,  or  a  secant 
plane  will  intersect  it  in  a  curve,  or  curves,  never  in  a 
straight  line. 

The  law  of  generation  of  a  douUe  curved  surface  is  not 
limited  to  the  motion  of  any  particular  curve  in  any  par- 
ticular surface,  for  the  curve  formed  by  any  secant  plane, 
can  be  conceived  of  as  one  which  could  generate  the 
surface,  by  moving  and  changing  its  shape,  according  to 
a  law.  The  best  definition  would  be  that  involving,  at 
once,  the  simplest  curve  of  section,  and  the  simplest  law 
of  motion,  to  be  decided  by  an  inspection  of  the  particular 
surface  in  question. 

158.  A  surface  of  revolution  is  one  generated  by  the 
revolution  of  a  line  about  a  right  line  as  an  axis.  The 
revolving  line  may  be  a  right  line  or  it  may  be  a  curve,  it 
may  cut  the  axis  or  it  may  not.* 

If  the  revolving  line  is  a  right  line,  it  generates  either 
the  cone,  when  it  intersects  the  axis  or  the  only  other 
surface  of  its  class,  known  as  a  hyperboloid  of  revolution. 
If  the  revolving  line  is  a  curve,  it  generates  a  double 
curved  surface,  and  has  no  straight  line  elements.  There 
may  be  an  infinite  variety  of  surfaces  of  revolution 
depending  upon  the  form  of  the  revolving  curve,  the 
simplest  being  the  following:  The  sphere,  formed  by  the 
revolution  of  a  circle  about  one  of  its  diameters;  a  cone, 
formed  by  the  revolution  of  a  right  line  about  another 
right  line  which  it  intersects;  a  cylinder,  formed  by  the 

*A  surface  of  revolution  Is  not  strictly  limited  to  one  formed  by  the  revolu- 
tion of  a  plane  curve,  although  In  general  such  are  the  ones  considered. 


SURFACES  OF  REVOLUTION  167 

revolution  of  a  right  line  about  another  right  line  to  which 
it  is  parallel;  an  ellipsoid,  formed  by  the  revolution  of  an 
ellipse  about  one  of  its  axes;*  a  paraboloid,  formed  by 
the  revolution  of  a  parabola  about  its  axis ;  a  Jiyperboloid 
of  revolution  of  two  nappes,  formed  by  the  revolution  of  a' 
hyperbola  about  its  transverse  axis,  etc.,  the  surface,  in 
general,  deriving  its  name  from  that  of  a  curve  which 
revolves. 

It  is  obvious  that  any  plane,  perpendicular  to  the  axis 
of  a  surface  of  revolution,  will  cut  circles  from  the  surface 
whose  centers  are  in  the  axis,  therefore,  it  is  possible  to 
conceive  of  such  a  surface  being  generated  by  the  motion 
of  a  circle,  such  that  the  center  moves  along  the  axis  of 
the  plane  of  the  circle  while  the  latter  changes  its  shape 
according  to  a  law. 

If  planes  are  passed  through  the  axis,  they  will  cut 
equal  curves  from  the  surface,  known  as  meridian  curves. 
That  meridian  curve  which  is  parallel  to  a  coordinate  plane, 
the  axis  being  also  parallel  to  the  coordinate  plane,  is 
known  as  the  principal  meridian  curve  A 

In  any  surface  of  revolution,  which  is  re-entrant,  the 
path  traced  by  the  point  of  the  revolving  curve,  nearest 
the  axis,  is  known  as  the  circle  of  the  gorge,  and  the  plane 
of  this  circle,  as  the  gorge  plane. 

159.  To  assume  a  point  upon  a  surface  of  revolution 
and  to  draw  a  tangent  plane  to  the  surface  at  the  point 

*  This  Is  sometimes  called  a  spheroid;  a  prolate  spheroid,  If  the  revolution  Is 
about  the  major  axis,  an  oblate  spheroid  if  the  revolution  is  about  the  minor 
axis. 

tin  discussing  surfaces  of  revolution,  let  it  be  understood  that  the  axis  of  the 
surface  is  taken  parallel  to  one  of  the  coordinate  planes  and  perpendicular  to 
the  other,  which  is  the  most  convenient  method  of  representation  for  the 
purpose  of  analysis. 


168  DESCRIPTIVE    GEOMETRY 

Analysis:— T/^e  point  will  lie  on  a  meridian  curve  of  the 
surface,  which  can  he  drawn,  and  a  tangent  plane  to  the 
surface,  at  the  point,  will  contain  a  tangent  to  the  meridian 
curve,  through  the  point.  Therefore,  draw  the  meridian 
plane  through  the  point,  and  if  it  is  not  the  principal 
meridian  plane,  revolve  it,  and  the  point  about  the  axis, 
until  it  becomes  the  principal  meridian  plane.  Its  projec- 
tion, upon  one  coordinate  plane,  will  lie  upon  the  contour  of 
the  surface.  Revolve  the  point  bach  again  into  its  original 
position;  the  projection  of  the  path,  on  one  coordinate  plane, 
will  be  a  circle,  that  on  the  other  will  be  a  parallel  to  the 
G,L, 

The  tangent  plane  contains,  in  addition  to  the  tangent 
to  the  meridian  curve,  through  the  point,  the  tangent  to 
the  circular  path  of  revolution  of  the  point  of  tangency, 
which,  in  turn  is  parallel  to  one  of  the  coordinate  planes; 
and  therefore,  one  trace  of  the  tangent  plane  will  be  par- 
allel to  one  projection  of  this  line. 

Construction:— Let  the  surface  be  given  as  an  oblate 
spheriod,  shown  as  in  Fig.  77  with  axis  parallel  to  Y  and 
perpendicular  to  H.  Either  projection  of  the  point  can  be 
assumed  as  a'.  Its  path  of  revoluton,  into  the  principal 
meridian  plane,  is  a  horizontal  line  in  V  projection,  and  a' 
falls  at  a^' ,  and  its  H  projection  at  a^  ,  Revolving  the 
point  back  again  to  its  originally  assumed  position,  a  i 
moves  in  the  arc  of  a  circle,  with  o  as  a  center,  to  the  posi- 
tion a,  on  a  perpendicular  through  a' ,  a',  as  assumed, 
could  stand  for  two  positions  of  the  point  on  the  surface, 
the  other  being  0^2 .  The  tangent  to  the  surface  at  A  can 
be  drawn,  at  the  revolved  position  of  A,  the  revolved  V 
projection  of  its  H  trace  being  bx' .    The  H  trace,  B,  re- 


SURFACES  OF  REVOLUTION 


169 


FlQUBB   77. 


volves  back  to  the  original  position,  and  falls  at  &'  in  V 
projection;  and  &  in  H  projection,  and  the  H  trace  of  the 
required  plane,  T,  is  tangent  to  the  circle  of  the  path  of  B, 
at  6.  The  V  trace  is  obtained  in  the  usual  manner,  or  it 
goes  through  the  V  trace  of  the  tangent  to  the  surface  at  A. 
The  tangent  to  the  meridian  curve,  in  revolving  from 
the  one  position  to  the  other,  generates  a  portion  of  the 
surface  of  a  right  circular  cone,  whose  apex  is  in  the  axis 
and  its  line  of  tangency  with  the  surface  may  be  con- 
sidered its  base. 


170  DESCRIPTIVE    GEOMETRY 

160  THEOREM  XV. 

A  plane  which  is  tangent  to  a  surface  of  revolution  at  a 

given  point  is  perpendicular  to  the  meridian  plane  through 

the  point. 

Proof;— The  tangent  plane  contains  the  tangent  to  the 
circle  of  revolution  of  the  point.  The  tangent,  being 
perpendicular  to  the  radius  of  this  circle,  is  also  perpen- 
dicular to  any  line  connecting  the  point  of  tangency  with 
the  axis  of  the  surface,  since  the  radius  is  perpendicular 
to  this  axis.  But  the  radius  and  the  axis  lie  in  a  plane, 
hence  the  tangent  is  perpendicular  to  this  plane  which  is 
the  meridian  plane. 

161.    To  pass  a  plane  tangent  to  a  sphere,  and  through  a 
given  line  outside. 

Analysis — TJie  given  line  will  he  the  intersection  of  the 
two  possible  planes  tangent  to  the  sphere,  through  the  line. 
Each  of  these  planes  will  he  perpendicular  to  a  radius  of 
the  surface  through  the  point  of  tangency.  The  plane  of 
these  normals  will  he  perpendicular  to  the  required  planes. 
Therefore,  pass  a  plane  through  the  center  of  the  sphere, 
and  perpendicular  to  the  given  line.  From  the  point  in 
which  it  is  pierced  hy  the  line,  draw  a  tangent  to  the  great 
circle  cut  from  the  sphere  hy  the  plane.  This  line,  and  the 
given  line,  constitute  one  of  the  possible  tangent  planes. 

Construction:— Let  the  sphere  be  given,  as  in  Fig.  78, 
and  the  line  as  AB.  Pass  a  plane  through  the  center  of 
the  sphere,  and  perpendicular  to  the  line  AB  by  using  a 
horizontal  of  the  plane  through  O.  The  line  AB  pierces 
this  auxiliary  plane  in  the  point  P.  Revolving  the  center 
of  the  sphere  O  and  the  point  P  into  H,  P  falls  at  p^  .    One 


SURFACES  OF  REVOLUTION 


171 


Figure    78. 

tangent  to  the  great  circle  cut  from  the  sphere,  by  the  aux- 
iliary plane,  is  Pi  r,  the  other  is  Pi  q.  Hence,  the  plane  of 
AB  and  PR,  in  the  one  case,  and  AB  and  PQ  in  the  other, 
determine  the  required  tangent  planes,  E  and  T,  respect- 
ively. 


162.  THEOREM  XVI. 

If  two  surfaces  of  revolution,  having  a  common  axis,  are 


172  DESCRIPTIVE    GEOMETRY 

tangenMo  each  other,  or  intersect,  it  will  be  in  the  circum- 
ference of  a  circle,  whose  plane  is  perpendicular  to  the  axis, 
and  center  in  the  axis. 

Proof:— If  a  plane  is  passed  through  any  point  of  the 
curve  of  intersection,  and  the  common  axis  of  the  sur- 
faces, it  will  cut  a  meridian  curve  from  each  surface, 
intersecting  each  other  in  a  point.  If  each  meridian 
curve  is  revolved  about  the  axis,  it  will  generate  the 
surface  to  which  it  belongs,  while  the  point,  common 
to  both  the  meridian  curves,  will  describe  the  circum- 
ference of  a  circle,  common  to  both  surfaces,  with  center 
in  the  axis,  and  plane  perpendicular  to  the  axis. 

163.    To  find  the  curve  of  intersection  of  any  surface  of 
revolution  by  an  oblique  plane. 

Analysis: — Planes ,  passed  through  the  axis  of  the 
surface,  will  cut  meridian  curves  from  it,  and  right  lines 
from  the  secant  plane,  the  points  of  intersection  of  which 
lie  on  the  curve  of  intersection  of  the  surface  and  the  plane. 
Or  again,  planes  perpendicular  to  the  axis,  will  cut  circles 
from  the  surface  and  lines  from  the  secant-plane. 

In  construction,  the  latter  are  to  be  preferred  because 
the  circles  are  easily  drawn,  if  the  axis  of  the  surface  is 
perpendicular  to  a  coordinate  plane,  and  the  lines  cut 
from  the  secant  plane  will  be  horizontals  or  verticals  of  the 
plane. 

Construction:— Let  the  surface  be  a  torus  form,  a  name 
derived  from  its  use  in  architecture,  a  surface  formed  by 
the  revolution  of  a  circle  about  a  line  lying  in  its  plane, 
but  not  passing  through  the  center.  It  is  shown  in  Fig. 
79,  and  the  secant  plane  as  T. 


SURFACES  OF   REVOLUTION 


173 


FlQUBK    79. 

Pass  a  series  of  secant  planes  perpendicular  to  V,  and 
to  the  axis  of  the  surface,  as  shown  at  1,  2,  3,  etc.  Their  V 
traces  will  be  parallel  to  the  G.  L.,  and  the  horizontals  cut, 
thereby,  from  the  plane  T  will  intersect  the  respective 
circles,  cut  from  the  surface,  in  points  of  the  curve  of  inter- 
section. The  V  traces  of  these  lines  are  identical  with  the 
traces  of  the  secant  planes  respectively. 


164.    The  hyperboloid  of  one  nappe,  or  hyperboloid  of  rev- 


174  DESCRIPTIVE     GEOMETRY 

olution,  as  it  is  called,  is  a  surface  generated  by  a  straight 
line,  which  moves  so  as  to  touch  three  windschief  lines, 
which  are  equi-distant  from  a  fourth,  and  make  equal 
angles  with  a  plane,  perpendicular  to  the  fourth.  Or  it 
may  be  considered  as  having  two  rectilinear  directrices, 
and  a  right  circular  cone  directer. 

This  surface  is  distinguished  from  the  h3rperbolic  para- 
boloid by  the  conditions  that  the  three  windschief  direct- 
rices shall  be  equally  distant  from  a  fourth,  and  shall  make 
equal  angles  with  a  plane  perpendicular  to  the  fourth. 

Its  construction,  however,  is  best  understood  when 
made  according  to  another  statement  of  the  law  of  genera- 
tion, namely,  a  surface  generated  by  a  line  revolving 
about  another  line,  which  it  does  not  intersect.* 

Let  Fig.  80  represent  a  hyperboloid  of  revolution,  con- 
structed as  follows :  —  Let  the  axis  of  revolution,  OP,  be 
perpendicular  to  H,  and  AB  be  a  limited  portion  of  the  re- 
volving line,  shown  here  as  parallel  to  V,  and  moving 
around  OP,  in  the  direction  of  the  arrow.  Its  perpendic- 
ular distance  from  the  axis  is  ep.  Every  point  of  the  line  AB 
describes  a  circle,  whose  center  is  in  the  axis,  and  whose 
plane  is  perpendicular  to  the  axis;  hence,  when  any  point, 
as  Gr,  has  revolved  to  a  position  G  i ,  in  a  plane  with  the 
axis  OP,  which  is  parallel  to  V,  it  will  be  a  point  in  the 
contour  of  the  surface ;  for  the  contour  is  the  line  of  tan- 
gency  of  a  cylinder  of  projection,  and  since  the  surface 
is  one  of  revolution  the  contour  is  in  a  plane  parallel  to  Y. 
All  points  of  the  line  AB  will  take,  successively,  positions 
in  the  principal  meridian  plane,  in  the  revolution  about  the 
axis,  and  other  points  in  the  contour  may  thus  be  plotted. 

*A  law  shortly  to  be  proved. 


SURFACES  OF    REVOLUTION 


175 


F  IGUBB    80. 


This  surface  is  symmetrical  on  its  axis,  so  that  for 
every  point,  as  G,  there  will  be  one  symmetrical  with  it 
upon  the  opposite  side  of  the  axis,  and  the  Y  projection, 
in  this  case,  will  be  symmetrical  with  respect  to  the  line 
o'p' ,  For  convenience,  in  the  illustration  chosen,  AB  is 
bisected  at  E,  which  describes  the  smallest  circle  of  revolu- 
tion of  any  point  of  the  line;  it  is  the  circle  of  the  gorge y 
and  its  plane  is  the  gorge  plane. 


176 


DESCRIPTIVE    GEOMETRY 


165.  THEOREM  XVII. 

The  hyperboloid  of  revolution  of  one  nappe  has  two 
systems  of  generation,  and  every  element  of  the  one  system 
intersects  all  those  of  the  other  system. 


/ 


A  \ 


a'j 

\               \     '"'      \ 

/ 

F  I  QUBE    81. 

Proofi — If  any  point  of  an  element  AB  as  E  in  Fig. 
83.,  revolve  through  an  angle  of  180°,  it  will  occupy  the 


SURFACES  OF  REVOLUTION  177 

position  F,  and  the  element,  that  of  the  position  of  CD, 
making  the  same  angle,  with  the  plane  perpendicular  to 
the  axis,  as  AB  does.  If,  now,  through  the  point  F, 
another  line  KL  is  drawn  making  the  same  angle  with 
this  plane,  and  lying  in  a  plane  parallel  to  V,  it  will,  if 
revolved  about  the  axis,  generate  the  same  surface,  for, 
any  plane  passed  perpendicular  to  the  axis,  will  cut  both 
these  elements  in  points,  respectively,  G  and  M  which 
are  equally  distant  from  the  axis,  and  these  each,  in 
revolution,  will  describe  the  same  circle  perpendicular  to 
the  axis,  and  so  for  any  other  piercing  points  of  the 
elements  with  any  plane  perpendicular  to  the  axis; 
hence,  the  surfaces  generated,  by  the  elements,  will  be 
identical.  And,  since  this  is  true,  through  any  point  of 
the  surface,  two  right  lines  can  be  drawn,  which  are 
elements  of  the  two  systems  of  generation,  respectively. 

Since  the  point  G,  of  the  element  AB,  and  the  point 
M,  of  the  element  KL,  generate  the  same  circumference, 
it  follows  that,  at  some  position  of  the  element  AB,  the 
point  G  coincides  with  the  point  M.  This  position  is 
shown  by  the  dotted  line  AiBj,  and  so  for  any  other 
point  of  AB.  Hence,  if  either  generatrix  remain  fixed, 
the  other  one,  in  revolution,  would  intersect  it  in  all 
points  successively.* 

And,  if  any  three  generatrices  of  one  system,  are 
taken  as  directrices,  and  a  generatrix  of  the  other 
system  moves,  so  as  to  touch  the  three,  it  will  generate 
the  surface,  hence  the  definition. 


*Thl8  property  Is  one  characteristic  of  warped  surfaces  In  general  bavins 
double  sj'stems  of  generation. 


178 


DESCEIPTIVE     GEOMETRY 


166.  THEOREM  XVIII. 

The  meridian  curve  of  a  hyperboloid  of  revolution  of  one 
nappe  is  a  hyperbola. 


Figure  88 . 

Proofi— Let  Fig.  83  be  a  hyperboloid  with  G  and  N, 
any  two  points  upon  it.  The  horizontal  distance  of  n^' 
from^'  is  less  than  that  of  g^'  from  g\  Still  further 
removed  from   G,  the   corresponding  projection   of  the 


SURFACES  OF  REVOLUTION  179 

distance  of  any  point,  on  the  meridian  curve,  from  the 
projection  of  the  generatrix,  when  parallel  to  one  coordi- 
nate plane,  is  less.  This  distance,  in  any  case,  as  g/g', 
is  the  difference  between  the  radial  distance  of  the 
point  from  the  axis,  sls  g  o,  and  its  distance  in  H  pro- 
jection from  the  point  of  tangency  of  the  element  AB, 
to  the  circle  of  the  gorge.  The  first  distance  is  the 
hypotenuse  of  a  right  angled  triangle  of  which  the 
second  distance  is  the  base,  and  the  distance  of  the  point 
of  tangency,  in  H  projection,  from  the  axis  is  the  al- 
titude. The  hypotenuse,  and  the  base  vary,  for  every 
point  on  the  element,  while  the  altitude  remains  con- 
stant. Let  the  hypotenuse  be  represented  by  h,  and  the 
base  by  h,  and  the  altitude  by  a.  Then  the  conditions 
may  be  expressed  algebraically,  as 
a'  =  }i-  —  y 

0Ta'=  (h-{-h)  ih-b) 
whence 

The  left  hand  member  diminishes  in  value  as  the 
denominator  of  the  right  hand  member  increases,  and 
becomes  equal  to  zero  when  the  denominator  becomes 
infinite.  Therefore,  the  element  AB  is  an  asymptote  to 
the  curve.  Again,  when  h  ==  zero,  h^  =  a%  and  the 
radius  of  the  circle  of  the  gorge  is  equal  to  a,  the 
nearest  point  of  the  moving  element  to  the  axis. 

Referring  also  to  Fig.  80,  and  remembering  that  the 
line  through  a  b  may  stand  for  the  H  projection  of  an 
element  of  each  system,  assume  any  other  position  of 
AB  as  AiBi.  It  will  cut  one  element  of  the  other 
system  in  a  point  M.,  and  a  second  element  in  a  point  N, 


180  DESCRIPTIVE    GEOMETRY 

on  the  opposite  side  of  the  circle  of  the  gorge.  Now, 
since  this  element  has  but  one  point  in  common  with 
the  meridian  curve,  the  vertical  projection  of  AiBi  will 
be  tangent  to  this  curve  at  K.  But  n  k  =  k  m,  then 
m'  ¥  =  ¥  n' ,  that  is,  the  tangent  to  the  meridian 
curve  is  bisected  at  the  point  of  contact,  a  well  known 
property  of-  the  hyperbola.  Hence,  the  meridian  curve 
is  a  hyperbola;  in  the  figure,  it  has  a  horizontal  trans- 
verse axis  and  a  vertical  conjugate  axis. 

The  surface  could  be  generated,  therefore,  by  revolu- 
tion of  a  hyperbola  about  its  conjugate  axis. 

If  the  circle  of  the  gorge  diminishes  in  diameter  to  zero, 
the  generatrices  will  go  through  its  center  and  the  surface 
becomes  a  cone. 

If  the  angle  of  the  generatrices  with  the  plane  directer 
diminishes  to  zero  the  surface  become  a  plane. 

167.  To  assume  a  point  on  the  surface  of  a  hyper- 
boloid  of  revolution  of  one  nappe,  and  to  draw  a  plane  tan- 
gent to  the  surface  at  the  point 

Analysis:— T/^e  elements  of  both  systems  of  generation, 
which  pass  through  the  assumed  point,  will  each  he  tangent 
to  the  projection  of  the  circle  of  the  gorge,  on  a  plane  per- 
pendicular to  the  axis  of  the  surface,  and  intersect  any 
assumed  bases  of  the  surface  in  points,  which  can  he  found. 
And  the  tangent  plane  will  contain  the  elements  of  the 
surface — one  of  each  system— passing  through  the  point  of 
contact,  will  he  tangent  to  the  surface  at  a  point,  and  cut  the 
surface  in  two  straight  lines. 


SURFACES   OF  REVOLUTION 


181 


Figure    83, 


Construction:— Let  Fig.  83  be  a  hyperboloid  of  revolu- 
tion. 

First,  assume  P,  in  H  projection,  in  which  case  either 
or  both  elements  can  be  drawn  through  it,  tangent  to  the 
circle  of  the  gorge,  and  cutting  the  base  in  the  circle  of 
contact  with  H,  from  which  their  V  projections  may  be  ob- 
tained. Or  the  V  projection  of  P  may  be  assumed.  In 
this  case,  revolve  the  point  about  the  axis  until  it  falls  into 
the  principal  meridian  plane,  and  after  its  H  projection 
has  been  obtained  in  this  position,  then  revolve  it  to  its 
original  position  and  obtain  the  true  H  projection. 

The  tangent  plane  contains  the  elements,  one  of  each 
system,  passing  through  the  point  of  contact,  audits  traces 


182  DESCRIPTIVE     GEOMETRY 

pass  through  the  traces  of  these  elements.    TV  and  TH  are 
the  traces  as  shown. 

168.    To   find   the   curve    of  intersection  of   the  hyper- 
boloid  of  revolution  with  any  plane. 

Analysis: — The  most  convenient  method  of  finding  the 
curve  of  intersection^  is  to  pass  a  series  of  auxiliary  secant 
planes  perpendicular  to  the  axis  of  the  surface,  cutting 
circles  from  the  hyperholoid,  and  right  lines  from  the  giveri 
plane,  the  respective  points  of  intersection,  of  which,  give 
points  upon  the  curve  of  intersection  of  the  two. 


CHAPTER   VIII 


INTERSECTIONS   OF  SURFACES. 

169.  Two  surfaces  will  intersect  each  other  in  curves  or 
straight  lines,  according  to  the  character  of  the  surfaces. 
And  the  method  of  ascertaining  the  form  of  the  curve,  will 
depend  also  upon  the  properties  of  the  surfaces  con- 
sidered; no  general  procedure  can  be  laid  down,  except 
the  following:  Pass  a  series  of  secant  planes  to  cut  from- 
both  surfaces,  the  simplest  lines  that  can  he  drawn,  the 
intersections  of  the  curves  cut  from  both  surfaces  by  the 
secant  planes,  respectively,  will  give  points  in  the  curve 
of  intersection  of  the  surfaces. 

The  subject  of  intersections  is  best  investigated  by- 
working  out  a  series  of  problems  involving  different  kinds 
of  surfaces.  The  intersection  of  the  simpler  single  curved 
surfaces,  and  surfaces  of  revolution  are  of  particular 
interest  to  the  student,  and  furthermore,  intersections, 
together  with  the  development  of  the  surfaces  intersected, 
form  a  class  of  problems  of  most  frequent  occurrence  in 
practical  descriptive  geometry,  and  deserve  to  be  dwelt 
upon. 

As  the  student  has  by  this  time  become  accustomed  to 
the  fundamental  processes,  it  is  proposed  to  curtail  the 
analyses,  somewhat,  and  to  cover  only  the  special  features 
involved  in  each  new  problem. 


384 


DESCRIPTIVE     GEOMETRY 


170.    To  find   the   intersection   of  two  bodies  bounded 
each  by  plane  faces. 

General  Analysis:  —  The  curve  of  intersection  will 
connect  the  points  of  intersection  of  the  various  edges  of  the 
one  form  with  the  surfaces  of  the  other  form.  Hence,  (i), 
find  the  various  lines  of  intersection,  of  the  planes  of  the  one 
form,  with  those  of  the  other,  respectively,  or,  (2),  find  the 
traces  of  the  edges  of  the  one  form  with  the  faces  of  the 
other,  the  latter  extended  if  necessary  to  contain  the  traces. 


Figure  84 . 


Construction:  —  Let  the    two  forms    be   two    prisms, 
square  and  triangular,  as  in  Figure  84.    Starting  with  any 


INTERSECTIONS  OF  SURFACES  185 

edge  of  one  of  the  prisms,  as  1,  of  the  square  prism,  see  if 
it  intersects  the  surfaces  of  the  triangular  prism,  by  using 
the  H  projecting  plane  of  the  edge,  and  finding  its  line  of 
intersection  with  the  faces  of  the  triangular  prism,  and  so 
on,  systematically,  each  edge  in  turn,  and  then  find, 
similarly,  the  piercing  points  of  the  edges  of  the  triangular 
with  the  square  prism,  wherever  the  intersection  has  not 
been  determined. 

171.  One  form    will  completely    penetrate    the    other 

when  two  parallel,  tangent  and  diametrically  opposite, 
planes  to  the  one  form  cut  lines  out  of  the  second  form. 
This  construction  is  very  useful  sometimes,  in  determining 
the  sphere  within  which  to  work  with  least  expenditure  of 
effort.  Two  sets  of  tangent  planes,  one  set  to  each  form, 
will  determine  at  once  the  limits  of  the  curve,  or  curves, 
of  intersection. 

To  make  the  statement  for  complete  or  partial  pene- 
tration general,  for  all  surfaces,  it  may  be  said  that  one 
completely  penetrates  the  other  when  two  tangent,  and 
opposite,  planes  to  the  one  form,  and  constructed  in 
accordance  with  the  limitations  imposed  hy  the  character  of 
the  surface,  cut  the  second  form. 

If  one  completely  penetrates  the  other,  there  will  be 
two  distinct  curves  of  intersection,  if  not,  the  curve  of 
intersection  will  be  single  and  continuous. 

172.  To  find  the  lines  of  intersection  of  two  pyramids. 
Analysis:— jPmc?  the  piercing  points,  of  the  edges  of  one, 

with  the  surfaces  of  the  other,  hy  means  of  auxiliary  planes, 
passed  through  the  edges  of  each,  and,  when  possible,  pass 
them  through  the  apexes  of  loth  pyramids;  whence,  the 


186  DESCKIPTIVE    GEOMETRY 

common  line  of  intersection,  of  the  several  secant  planes,  is 
the  line  connecting  the  apexes  of  the  two  pyramids,  and  the 
lines  cut  from  both  pyramids  are  elements  of  them  respec- 
tively, 

173.    To  find  the  curve  of  intersection  of  two  cylinders^ 

Analysis:— 7/  the  planes  of  the  bases  of  the  two  cylinders 
are  parallel,  choose  the  secant  planes  parallel  to  the  bases. 
Similar  curves  to  the  bases  will  be  cut  from  each,  and  their 
respective  intersections  give  points  on  the  curve  desired.  If 
the  bases  are  not  in  parallel  planes,  pass  secant  planes 
through  the  elements  of  one,  which  are  parallel  to  those  of 
the  other,  hence,  cutting  elements,  if  at  all,  out  of  both. 

Construction:— Let  two  cylinders  be  given,  as  in  Fig. 
85,  having  axes  AB  and  CD  respectively.  Since  the  second 
cylinder  has  its  axis  parallel  to  the  G.  L.,  any  auxiliary 
planes,  passed  parallel  to  its  elements,  will  be  parallel  to 
the  G.L.  Hence,  to  find  the  intersections  of  such  a  series 
of  auxiliary  planes  with  both  cylinders,  project  the  cylind- 
ers upon  and  end  plane  as  shown.  The  respective  inter- 
sections of  the  contour  of  both  cylinders,  with  any  one  aux- 
iliary plane,  are  the  piercing  points  of  elements  of  the 
cylinders,  with  their  respective  bases,  and  these  can  be 
transferred  to  the  coordinate  planes.  One  such  auxiliary 
plane  is  shown,  cutting  the  elements  EF  and  IJ,  respec- 
ively,  out  of  both  cylinders. 

Note.  In  the  practical  application  of  descriptive  ge- 
ometry, in  such  problems  as  these,  it  is  found  convenient 
to  choose  one  coordinate  plane  parallel  to  the  axes  of  both 
solids,  so  that  the  elements  cut  out  will  be  projected  their 
true  length  on  that  coordinate  plane,  since  the  main  object 


INTERSECTIONS  OF  SURFACES 


187 


Figure  85. 


of  ascertaining  the  intersection  is  that  the  forms  may  be 
developed,  such  for  example  as  two  intersecting  pipes. 


174.  To  find  the  curve  of  intersection  of  two  cones. 
Analysis:—  Unless  the  bases  are  circular  and  in  parallel 
planes,  pass  a  series  of  secant  planes  to  cut  elements  from 
both  cones.  These  planes  will  intersect  each  ether  in  a  line 
through  the  apexes  of  both  cones,  and  their  traces,  moreover, 
will  go  through  the  traces  of  this  line.  The  respective 
intersections  of  the  elements,  cut  from  the  cones,  will  be 
points  in  the  curve  of  intersection  sought. 


188 


DESCRIPTIVE  GEOMETRY 


Figure   86. 

Construction:— Let  two  cones  be  given,  as  in  Figure  86. 
The  circular  base  of  one  is  in  V,  with  apex  at  O,  and  the 
circular  base  of  the  other  is  in  H,  with  apex  at  C.  The  line 
OC,  connecting  the  apexes,  has  its  V  and  H  traces  in  A 
and  B,  respectively,  and  the  secant  planes  used  will  con- 
tain these  traces.  Planes  T  and  R,  drawn  tangent  to  the 
first  mentioned  cone,  show  that  T  cuts  elements  out  of  the 


INTERSECTIONS  OF  SURFACES  189 

second,  but  R  does  not,  likewise  planes  S  and  U,  drawn 
tangent  to  the  second  show  that  S  cuts  elements  out  of  the 
first,  but  U  does  not ;  hence,  neither  cone  completely  pen- 
etrates the  other.  The  construction  of  points  by  use  of 
another  auxiliary  plane  is  shown. 

If  both  bases  are  in  the  same  coordinate  plane,  only  one 
trace  of  a  secant  plane  is  necessary  to  determine  whether 
that  plane  cuts  both  cones.  Its  corresponding  trace  may 
be  found  with  a  horizontal,  or  vertical,  of  the  plane,  or  it 
may  not  be  needed  in  construction. 

If  both  cones  have  the  same  apex,  either  trace  of  any 
secant  plane  can  be  assumed  at  will,  the  corresponding 
trace  being  determined  by  the  fact  that  the  apex  is  a  point 
in  the  plane. 

If  the  line  connecting  the  apexes  of  the  cones,  is  parallel 
to  their  bases,  the  traces  of  the  auxiliary  planes,  with 
those  bases,  will  be  parallel  lines. 

175.    To  find  the  curve  of  intersection   of  a  cone   and 
a  cylinder. 

Analysis:— T/^e  secant  planes,  to  cut  elements  out  of 
each,  must  pass  through  the  apex  of  the  cone,  and  be  parallel 
to  the  elements  of  the  cylinder;  hence,  draw  a  parallel  line 
through  the  apex  of  the  cone,  as  the  common  line  of  inter- 
section of  the  secant  planes. 

Construction:— Let  the  cone  and  cylinder  be  given  as 
in  Figure  87.  Draw  a  line  through  the  apex  of  the  cone, 
as  the  line  of  intersection  of  the  secant  planes.  In  this 
case,  it  is  perpendicular  to  H.  The  secant  planes  T,  E,  S, 
etc.,  are,  therefore,  perpendicular  to  H,  and  cut  both 
surfaces  in  straight  lines.  The  H  traces  can  be  assumed 
at  will,  to  pass  through  the  H  trace  of  the  auxiliary  line, 


190 


DESCRIPTIVE   GEOMETRY 

to' 


FIGTJBE     87. 

through  the  apex  of  the  cone.  The  points  of  intersection, 
D  and  E,  of  the  elements  cut  out  of  each,  by  one  auxiliary- 
plane  R,  are  shown. 

Where  the  bases  of  cone  and  cylinder  are  in  the  same 
plane,  the  traces  of  the  secant  planes,  with  that  plane,  can 
be  assumed  at  will  to  go  through  the  trace  of  an  auxiliary 
line,  through  the  apex  of  the  cone,  and  parallel  to  the 
elements  of  the  cylinder.    If  this  auxiliary  line  is  parallel 


INTERSECTIONS  OF  SURFACES  191 

to  the  plane  of  the  bases,  the  traces  of  the  secant  planes, 
on  that  plane,  are  parallel. 

If  the  bases  of  cone  and  cylinder  are  in  or  parallel  to 
different  coordinate  planes,  then  either  of  the  traces  of  the 
auxiliary  planes  may  be  assumed  at  will,  to  go  through  the 
similar  trace  of  an  auxiliary  line,  through  the  apex  of  the 
cone,  and  parallel  to  the  elements  of  the  cylinder. 

The  proper  traces  to  assume,  of  the  auxiliary  planes, 
are  those  with  the  coordinate  plane  in  which  the  bases  re- 
spectively lie,  for  the  traces  show,  at  once,  the  elements 
cut  from  that  solid  whose  base  lies  in  that  plane. 

176.  To  find  the  curve  of  intersection  of  a  cylinder  and  a 
convolute. 

General  Analysis:— 7/"  the  base  of  the  cylinder  lies  in  or 
parallel  to  the  plane  of  section  of  the  convolute,  used  to 
represent  part  of  the  contour  of  the  latter,  then  secant 
planes,  parallel  to  the  base  of  the  cylinder,  will  probably 
give  the  simplest  construction,  although  cutting  curves  out 
of  both.  If  the  base  of  the  cylinder  is  not  so  related  to  the 
convolute,  then,  through  any  point  of  an  element  of  the 
latter,  draw  an  auxiliary  line  parallel  to  the  elements  of  the 
cylinder;  the  vlane  of  these  two  lines  determines  the 
direction  of  one  auxiliary  plane, 

177.  To  find  the  curve  of  intersection  of  a  cone  and  a 
convolute. 

General  Analysis: — If  the  base  of  the  cone  lies  in,  or 
parallel  to  the  plane  of  section  of  the  convolute,  used  to 
represent  part  of  the  contour  of  the  latter,  then  secant 
planes,  parallel  to  the  base  of  the  cone,  will  give  the 
simplest  construction,  although  cutting  curves  out  cf  both. 


192  DESCRIPTIVE  GEOMETRY 

If  the  base  of  the  cone  is  not  so  related  to  the  convolute,  the 
general  solution  involves  the  cutting  of  a  series  of  differing 
curves  from  the  convolute,  while  cutting  curves  or  elements 
from  the  cone. 

178.  Two  intersecting  surfaces  of  revolution  may  come 
under  one  of  three  groups.  1.  When  they  have  a  common 
axis.  2.  When  the  axes  are  in  the  same  plane.  3.  When 
the  axes  do  not  intersect. 

If  two  intersecting,  or  tangent,  surfaces  of  revolution 
have  a  common  axis,  they  will  touch  each  other  on  a  circle 
or  circles;  for,  having  a  common  axis,  any  point,  common 
to  the  two,  must  lie  on  a  meridian  curve,  and,  in  each  case, 
if  revolved,  will  generate  ihe  surface,  every  point  travel- 
ing in  the  path  of  a  circle,  whoes  center  is  in  a  plane  per- 
pendicular to  the  axis. 

179.  To  find  the  intersection  of  two  surfaces  of  revolu- 
tion having  a  common  axis. 

Analysis: — If  the  axis  is  taken  parallel  to  one  coordinate 
plane,  and  perpendicular  to  the  other,  then  secant  planes, 
parallel  to  the  latter,  will  cut  both  surfaces  in  circles,  which 
intersect  in  points  desired.  If  the  axis  is  not  so  placed,  a 
change  of  coordinate  plane  or  planes  would  be  most 
desirable, 

180.  To  find  the  curve  of  intersection  of  two  surfaces  of 
revolution  whose  axes  are  in  the  same  plane. 

Analysis:—//  the  axes  are  parallel,  pass  a  series  of 
auxiliary  planes  perpendicular  to  the  axes,  the  curves  cut 
from  each  surface  will  be  circles. 

If  the  axes  are  not  parallel,  the  convenient  method  is 


INTERSECTIONS  OF  SURFACES 


193 


to  intersect  both  surfaces  with  a  series  of  spheres,  of  dif- 
ferent radii,  and  with  centers  at  the  point  of  intersection 
of  the  axes.  These  spheres  will  cut  circles,  respectively, 
from  both  surfaces.  If  the  plane  of  the  axes  is  taken 
parallel  to  a  coordinate  plane,  the  projection  of  these 
circles  on  this  coordinate  plane  will  be  limited  and  inter- 
secting straight  lines,  the  points  of  intersection  of  which, 
respectively,  are  the  common  chords  of  the  circles  cut 
from  each  surface,  one  point  each,  as  a  common  chord  to 
the  two  circles  cut  from  the  surfaces  by  any  one  sphere. 


FIGUBE    88 


Construction: — Let  the  two  surfa'ces  be  an  ellipsoid  of 
revolution,  or  prolate  spheroid,  and  a  right  elliptical  cone, 
whose  apex  C,  is  on  the  surface  of  the  spheroid.  See 
Figure  88.    The  center  for  a  series  of  auxiliary  circles  is  at 


194  DESCBIPTIVE  GEOMETRY 

O,  the  intersection  of  the  axes  of  the  surfaces.  A  sphere, 
such  as  1,  and  another  such  as  3,  since  they  go  through 
the  points  of  intersection  of  the  contours  of  the  solids,  and 
since  the  plane  of  the  axes  is  parallel  to  V,  will  cut  out 
chords  of  zero  length,  i.e.,  be  the  piercing  points  of  those 
elements  of  the  cone  which  are  parallel  to  V.  Sphere  2 
will  cut  out  a  circle  from  each  intersecting  in  a  common 
chord  CD,  the  circle  cut  from  the  spheroid  is  vertically 
projected  in  the  line  i'g'  and  the  circle  cut  from  the  cone, 
is  vertically  projected  in  the  line  e'f.  In  like  manner 
other  points  are  ascertained. 

181.  To  find  the  curve  of  intersection  of  two  surfaces 
of  revolution  who  axis  are  not  in  the  same  plane. 

Analysis:— Pa55  secant  planes  to  cut  the  simplest  series 
of  lines  out  of  each  surface,  circles  out  of  one,  for  example, 
hy  planes  perpendicular  to  the  axis,  and  curves  out  of  the 
other. 

No  general  analysis  leading  to  a  shnple  solution  under 
these  conditions  is  possible. 

182.  To  find  the  curve  of  intersection  of  a  single 
curved  surface  and  a  surface  of  revolution. 

Analysis: — Pass  a  series  of  secant  planes  normal  to  the 
axis  of  the  surface  of  revolution,  to  cut  circles  from  it  and 
circles,  ellipses,  or  other  curves  from  the  other  surface. 

If  the  single  curved  surface  is  a  cone,  or  cylinder,  it 
is,  in  general,  possible  to  get  fairly  simple  curves  of  inter- 
section. If  the  surface  is  a  sphere,  also,  the  curves  of 
intersection  will  be  circles.  If  it  is  a  convolute  surface, 
the  curves  cut  out  will  depend  upon  the  position  of  the 
axis  of  the  convolute  relative  to  that  of  the  surface  of 
revolution. 


APPENDIX 


Practical  projection  is  based  upon  descriptive  geometry,  but  the 
short  cuts,  in  common  use,  are,  many  of  them,  entirely  at  variance 
with  it.  They  are  correct  in  their  way,  but  the  practical  and  the 
theoretical  should  not  be  confused.  For  example,  the  real  differ- 
ence between  first  and  third  angle  projection  has  not  always  been 
followed  by  draftsmen,  the  two  have  been  mixed.  In  the  first  angle, 
an  object  is  projected  upon  the  coordinate  planes  from  a  center  of 
projection,  which  is  on  the  same  side  of  the  plane  as  the  object;  in 
the  third  angle,  the  center  of  projection  is  on  the  opposite  side  of 
the  plane  from  the  object.  This,  in  the  case  of  the  first  angle, 
results  in  an  end  view  of  an  object  being  placed  upon  the  opposite 
side  from  the  one  the  view  is  supposed  to  represent,  while,  in  the 
third  angle,  the  end  view  comes  next  to  the  side  which  it  is 
supposed^to  represent.  Further  than  this,  in  the  first  angle,  the 
revolution  of  the  end  view  is  away  from  the  end  it  represents,  and 
in  the  third  angle  it  is  toward  it,  theoretically,  while  in  practical 
projection,  for  both  angles,  the  revolution  is  away  from  the  end 
it  represents,  thus,  theoretically,  dividing  the  end  plane,  on  its 
intersection  with  V,  and  folding  the  front  and  rear  parts  on  each 
other  like  the  leaves  of  a  book.  If  these  distinctions  are  kept  in 
mind,  the  student  will  not  be  likely  to  mix  the  first  and  third 
angles  in  his  practical  work.  Since  practical  projection  is  but  a 
language  constructed  to  give  expression  to  ideas,  it  should  be 
allowed  license  in  its  mode  of  expression,  to  best  suit  its  purposes, 
as  long  as  there  is  consistency,  and  a  common  understanding  and 
approval  of  what  is  meant. 

Practical  projection  ignores  the  ground  lines  between  any  coor- 
dinate planes  of  projection,  and  for  the  very  good  reason  that  they 
in  no  way  affect  the  projections  of  forms  or  the  relations  of  the 
projections  to  each  other  except  as  to  distance  apart;  and,  because 
ground  lines  are  ignored,  in  various  other  ways,  detail  views  are 
arbitrarily  placed  where  they  will  be  most  explanatory.  As  a  con- 
crete illustration  of  this,  the  section  of  a  spoke  of  a  wheel  may  be 


198  APPENDIX 

revolved  about  the  median  line  of  the  section,  where  taken,  and 
shown  as  limited  by  the  outlines  of  the  spoke.  Again,  it  may  be 
said,  that  such  arbitrary  constructions  are  permissible  if  well 
understood  and  accepted  by  custom. 

A  very  large  share  of  the  practical  problems  involving  descrip- 
tive geometry  are  those  of  ascertaining  the  direction  and  shape  of 
sections  of  forms,  and  of  the  development  of  the  forms.  Not  all 
peripheries  of  solids  can  be  developed,  in  the  true  sense  of  the 
word,  while,  in  reality,  all  forms  are  developed  in  practice.  The 
only  forms  that  can  be  developed,  in  conformity  with  the  princi- 
ples of  descriptive  geometry,  are»  those  having  straight  line 
elements  and  intersecting,  at  least  consecutively,  two  and  two. 
No  double  curved  or  warped  surfaces  are  developable. 

In  practical  work  these  latter  surfaces  frequently  must  be 
shaped  up,  notably  in  the  sheet  metal  worker's  business,  and 
interesting  devices  are  resorted  to.  For  double  curved  surfaces 
there  are  two  general  methods:  1.  The  zone.  2.  The  gusset. 
In  the  zone  method,  the  solid  is  divided  into  zones,  by  planes  per- 
pendicular to  a  principal  axis,  if  there  be  any,  and  the  zones  again 
broken  up  into  two  or  more  parts,  to  get  smaller  pieces,  more 
closely  approximating  the  real  form.  The  gusset  method  is  best 
illustrated  by  reference  to  the  periphery  of  an  orange  when 
divided  into  its  natural  lobes,  or  to  a  sphere,  divided  by  meridians 
into  the  necessary  number  of  slices,  closely  approximating  the 
true  form.  In  the  sheet  metal  worker's  business,  the  material  is 
sometimes  punched  out  or  stretched  to  make  it  fill  out  the  remain- 
ing difference  between  the  true  and  the  constructed  forms. 

Other  forms  are  developed  by  a  system  known  as  triangulation. 
The  surface  is  divided  up  into  triangles  having  their  correspond- 
ing bases  oppositely  directed.  In  the  development,  each  triangle 
is  revolved  about  its  edge  of  contact  with  an  adjoining  one,  and 
the  bases  of  the  alternate  triangles  form  a  continuous  develop- 
ment of  the  perimeter  of  a  portion  of  the  form;  opposite  to  these, 
the  bases  of  another  set  of  alternate  triangles  go  to  make  up 
another  portion  of  the  perimeter.  The  frustrum  of  an  oblique 
cone  is  a  surface  which  might  be  dealt  with  in  this  way,  but  the 
method  can  also  be  applied  to  other  surfaces. 


EXERCISES 


Following  Section  24,  Page  17. 

1.  Draw  the  projections  of  the  following  points:  A  =  0,  If,  |; 
B  =  7,  If,  —  i;  C  =  1\,  —  1,  —  i;  D  =  2^,  —  If,  i;  E  =  3i,  —  If,  0. 
Project  them  upon  a  new  Vi  plane  whose  Gi-L^.  cuts  the  G.L.  at  3i, 
0,  0,  and  makes  an  angle  of  30°  with  it  downward  towards  the  right. 

2.  Draw  the  projections  of  the  following  lines:  A  B  has  A  =  0, 
i^,  —  1^;  B  =  li,  1^,  —  li^.  C  D  has  C  =  2,  If,  f ;  the  line  pierces 
the  G.L.  and  d'  is  at  3t^,  —  f ,  0.  Project  both  of  these  lines  upon 
a  new  Vj  plane  whose  Gi.Lj.  is  parallel  to  the  H  projection  of  C  D 
and  cuts  the  G.L.  at  If,  0,  0. 

Following  Section  26,  Page  19. 

3.  Given  two  points  A  =  0,  li,  f ,  and  B  =  If ,  —  1\  — H;  draw 
the  projections  of  the  line  joining  them. 

4.  Given  two  points,  A  =  0,-11,  —  1,  and  B  =  2,  f ,  1|;  draw 
the  projections  of  the  line  joining  them. 

5.  A  B,  C  D  and  E  F  are  three  lines.  A  =  0,  0,  0,  and  B  =  U, 
if,  1^;  C  =  U,  -  f,  -  f,  and  D  =  2t,  -  f,  -  1;  E  =  3,  -  Ih  -  f, 
and  F  =  4,  0,  —  f .  What  are  the  relations  of  these  lines  to  the 
coordinate  planes?  the  angles  CD  and  EF  with  them? 

6.  A  B  and  C  D  are  two  lines.  A  =  0,  2^,  li^,  and  B  =  U,  1,  i^; 
C  =  f ,  1, 1,  and  D  =  If,  If,  H.    Draw  their  projections. 

7.  A  B  is  a  line  with  A  =  0,  f ,  U,  and  B  =  If,  U,  i.  C  and  D 
are  points  on  the  line;  C  is  in  a  vertical  plane  j  inch  to  the  right 
of  A,  and  D  lies  in  a  horizontal  plane  f  inch  below  B.  Locate 
both  projections  of  the  points. 

8.  A  B  is  a  line,  with  A  =  0,—  U,  —  U,  and  B  =  H  —  i,  —  |.  C 
is  at  If,  —  1,  —  If.    Through  C,  draw  any  line  to  intersect  A  B. 


200  EXERCISES 

9.  A  B  is  a  line,  with  A  =  0,  1|,  —  |,  and  B  =  U,  If,  —  |.  C  is 
the  middle  point  of  A  B.  D  is  a  point  at  i,  i,  f .  Through  D,  draw 
a  line  to  intersect  the  line  A  B,  in  the  point  C. 

10.  A  B  is  a  line,  with  A  =  0,  i,  U,  and  B  =  If,  U,  i.  C  is  at 
I4,  i,  If-  From  C,  as  one  extremity,  draw  a  line  so  that  its  middle 
point  shall  lie  on  the  line  A  B. 

11.  Locate  the  projection,  on  a  right  hand  end  plane,  of  a  line 
whose  one  end  is  }  inch  from  V,  and  f  inch  from  H,  in  the  first 
angle,  and  whose  other  end  is  i  inch  from  H,  and  f  inch  from  V, 
in  the  third  angle.  The  ends  of  the  line  are  H  inches  apart  hori- 
zontally. Draw  the  projections  of  the  line  in  either  of  its  two 
possible  positions  upon  the  H  and  V  planes. 

12.  Draw  the  projections  upon  the  H,  V  and  Ve  planes  of  the 
two  lines  joining  the  points  AB  and  CD.  A  =  0,  i,  f,  and  B  =  2,  H, 
1;  C  =  i,  i,  —  If,  and  D  =  If,  —  1,  f.  Let  Ge.Le.  cut  the  G.L. 
at  3f ,  0,  0. 

13.  Draw  the  projections,  upon  the  H,  V  and  Ve  planes,  of  the 
two  lines  joining  the  points  A  B  and  CD.  A  =  0,  H,  —  I,  and 
B  =  0,  —  t,  U;  C  =  i,  —  If,  —  1,  and  D  =  2,  0,  0;  Ge.Le.  cuts  the 
G.L.  at  3.  0,  0. 

14.  Given  the  line  A  B,  with  A  =  0,  li,  |,  and  B  =  If,  i.  If;  draw 
the  projecting  planes  of  the  line. 

15.  A  point  A  =  0,  f ,  i  and  a  point  B  =  0,  U,  1;  find  the  projec- 
jections  of  a  line  C  D,  2  inches  long  and  parallel  to  the  line  A  B, 
and  distance  f  inch  from  it,  in  a  vertical  plane  through  the  line 
AB. 

16.  Draw  the  lines  A  B  and  C  D  which  intersect  each  other  in 
a  point  O,  at  0,  f ,  —  1  i.  A  B  is  parallel  to  V  and  oblique  to  H, 
and  C  D  is  parallel  to  H  and  oblique  to  V.  Coordinate  the 
extremities  of  the  limited  line  used. 

17.  Given  the  line  A  B  C,  B  being  the  middle  point  in  it.    A 


EXERCISES  201 

=  0,  U,  I,  and  C  =  0,  i,  1\.    Draw  a  line  through  B  parallel  to  a 
line  E  F,  in  which  E  =  f ,  i,  i,  and  F  =  If,  |,  f . 

18.  Given  a  line  A  B;  A  =  0,  f,  |,  and  B  =  l-J,  If,  1.  Draw 
limited  lines  parallel  to  it  lying  in  each  of  the  four  dihedral  angles. 

19.  Prove  that  the  lines  of  problem  12  do  not  intersect. 

To  Follow  Section  32,  Page  24. 

20.  Find  the  traces  of  the  lines  A  B  and  CD.  A  =  0,  H,  —  |, 
and  B  =  li,  -  f ,  li;  C  =  If,  If,  i,  and  D  =  3f ,  f ,  U. 

21.  Find  the  traces  of  the  line  A   B.    A  =  0,  —  |,  —  If ;  B  =  2, 

16»   T6« 

22.  Find  the  traces  of  the  line  M  N.  M  =  0,  U,  U;  N  =r  2f ,  0,  0. 
Also  find  its  traces  with  an  end  plane  Ge.Le.  cutting  G.L.  at  If, 
0,  0,  showing  same  as  revolved  into  V  about  Ge.Le  . 

23.  A  B  is  a  line  with  A  =  0,  li,  —  U;  B  =  1|,  i,  —  \.  Locate 
its  traces  with  V  and  H  and  also  with  an  end  plane  whose  Ge-Lb. 
cuts  G.L.  at  1,  0,  0,  revolving  the  same  into  the  V  plane. 

24.  Find  the  traces  of  the  broken  line  connecting  successively 
the  points.    A  =  0,  -  f ,  -  i^;  B  =  U,  H,  -  i;  C  =  2f,  f ,  If;  D  = 

To  Follow  Section  33,  Page  25. 

25.  Draw  a  line  A  B,  with  its  V  trace  at  0,  f ,  0,  which  passes 
through  the  1st  and  2nd  angles  only. 

26.  Draw  a  line  A  B  with  its  V  trace  at  0,  |,  0,  which  passes 
through  the  2nd,  1st,  and  4th  angles. 

27.  Given  a  line  A  B  with  A  =  0,  i,  —  i;  B  =1,  U,  —  1\.  What 
is  the  position  of  the  line  with  respect  to  H  and  V;  where  are  its 
traces? 

28.  Given  a  line  A  B,  with  A  =  0.  If,  —  i;  B  =  U,  i  -  If. 
Find  its  traces  and  also  its  traces  with  an  end  plane  whose  Ge.Le 


202  EXERCISES 

passes  through  the  point  of  intersection  of  the  V  and  H  projections 
of  AB. 

29.  The  V  trace  of  a  finite  line  A  B  is  U  inches  below  H  and  its 
H  trace  is  H  inches  back  of  V.  Draw  the  projections  of  such  a 
line  and  find  its  traces  with  an  end  plane  whose  GeXe.  cuts 
through  the  middle  of  the  line. 

To  Follow  Section  38,  Page  32. 

30.  A  B  is  a  line,  with  A  =  0,  i,  i;  B  =  1|,  U,  H.  Find  the  true 
length  of  the  line,  by  revolving  it  about  the  horizontal  projecting 
line  of  the  point  B  until  it  is  parallel  to  V.  Show,  by  appropriate 
notation,  the  H  and  V  projections  of  the  arc  of  revolution  of  the 
point  A. 

31.  A  B  is  a  line,  with  A  =  0,  |,  U;  B  =  U,  If,  |.  Find  the  true 
length  of  the  line  by  revolving  it  about  the  vertical  projecting  line 
of  the  point  B  until  it  is  parallel  to  H.  Show,  by  appropriate 
notation,  the  H  and  V  projections  of  the  arc  of  revolution  of  the 
point  A. 

32.  A  B  is  a  line,  with  A  =  0,  H,  0;  B  =  H,  0,  f.  Find  the  true 
length  of  the  line,  by  revolving  it  about  the  H  trace  of  its  H  pro- 
jecting plane  into  H.  Also,  find  its  length,  by  revolving  it  about 
the  V  trace  of  its  V  projecting  plane  into  V. 

33.  AB  is  a  line,  with  A  =  0,  If,  f ;  B  =  li,  f,^.  Find  the  true 
length  of  the  line,  by  projecting  it  upon  a  supplementary  Vi  plane, 
taken  in  any  convenient  place,  parallel  to  AB. 

34.  A  line  goes  through  the  points  A  and  B  with  A  =  0,  i,  —  i, 
B  =  1,  H,  —  H.  Locate  any  two  points,  O  and  P,  on  it  which  are 
li  inches  apart. 

35.  Given  the  four  points.  A,  B,  C,  and  D.  A  =  0,  i,  |;  B  =  If, 
U,  H;  C  =  i^.lf,  If ;  D  =  2,  0,  i.  Find  the  distances  between  the 
extremities  of  the  two  lines  AB  and  CD.  Also,  find  the  distances 
between  P,  on  AB,  |  inch  from  A,  and  Q,  on  CD,  i  inch  from  C. 

36.  A,  the  V  trace  of  a  line  AB,  is  at  0,  —  2i,  0;  B  the  H  trace 


EXERCISES  203 

of  the  line  is  at  2|,  0,  —  2^;  C,  the  H  trace  of  another  line  inter- 
secting AB,  is  at  1^,  0,  —  If;  D,  the  point  of  intersection  has 
its  z  =  li.  Draw  the  lines  AB  and  CD,  and  find  the  V  trace  of 
CD.  Through  D,  draw  a  line  whose  V  trace  has  z  =  f ,  and  is  i 
inch  below  H. 

37.  A  =  0,  f,  i;  B  =  I,  If,  U.  Find  the  true  length  of  the  line 
by  the  following  methods:  (a)  by  revolving  parallel  to  V;  (b),  by 
revolving  parallel  to  H;  (c),  by  revolving  about  its  H  projection 
into  H;  (d),  by  revolving  about  its  V  projection  into  V;  (e),  by 
projecting  upon  a  new  Vi  plane  parallel  to  the  line,  using  a  Gi.  Li 
to  cut  the  G.L.  at  |,  0,  0. 

38.  Find  the  true  length  of  the  line  AB  with  A  =  0,  U,  —  U; 
B  =  li,  0,  0. 

To  Follow  Section  46,  Page  38. 

Where  TO  is  given  it  means  the  point  in  which  the  plane  T  cuts 
the  G.L. 

Point  1  stands  for  a  point  of  a  plane  in  V,  2,  for  a  point  of  the 
plane  in  H,  i,  e.,  points  on  the  traces. 

39.  T  is  a  plane  with  TO  at  0,  0,  0.  TH  makes  an  angle  of  30° 
with  the  G.L.,  and  TV  makes  an  angle  at  60°  with  the  G.L.  both 
towards  the  right.  A,  B,  C  and  D  are  four  points  in  the  plane  T, 
and  their  vertical  projections  are  as  [follows:  a'  =l,f,  0;  b'  = 
—  i,  —  I,  0;  c'  =  i,  —  H,  0;  d'  =0,  H,  0.  Find  the  H  projections 
of  the  points  by  using  a  horizontal  of  the  plane,  in  each  case. 

40.  T  is  a  plane  with  TO  at  0,  0,  0.  TH  makes  an  angle,  with 
G.L.  whose  tangent  is  f ;  TV  makes  an  angle  with  G.L.  whose 
tangent  is  |,  both  towards  the  right.  A,  B,  C  and  D  are  four 
points  in  the  plane  and  their  horizontal  projections  are  respec- 
tively as  follows:  a  =  2,  0,  |;  6  =  1,  0,  — 1;  c  =  —  i,  0,  —  i,  and  d 
=  H,  0,  0.  Find  the  vertical  projections  of  the  points  by  using  a 
vertical  of  the  plane  in  each  case. 

41.  Given  the  plane  T,  with  TO  =  0,  0,  0.  TH  has  2  at  2,  0,  If; 
TV  has  1  at  2,  li,  0;  apoint  O,  which  is  in  the  plane,  has  o  =  li,  0,i. 
Draw  any  two  lines  through  the  point  O  and  lying  in  the  plane  T. 


204  EXERCISES 

42.  Given  the  three  points,  A  =  0,  i,  i^;  B  =  2i,  0,  2,  and  C  = 
¥,  1, 1^;  to  pass  a  plane  through  them. 

43.  Given  two  parallel  lines  AB  and  CD;  A  =  0,  |,  i;  B  =  f,  |, 
I;  C  =  i,  f ,  1.     Pass  a  plane  through  the  two  lines. 

Note. — Parallel  lines  have  parallel  projections. 

44.  The  line  AB  has  A  =  0,  i,  |;  B  =  |,  ^,  i.  Pass  a  plane 
through  the  line  AB  and  a  point  C  =  If,  1,  —  |. 

45.  The  line  AB  has  A  =  0,  U,  i;  B  =  If,  —  i,  U.  Pass  a  plane 
through  the  line  AB  and  a  point  C  =  If,  li,  —  i. 

46.  Given  the  vertices  A,  B  and  C  of  a  triangle:  A  =  0,  f ,  li; 
B  =  f ,  li,  f ;  C  =  If,  i,  1.  Find  the  lengths  of  the  edges  of  the 
triangle  by  revolving  it  into  H  about  the  H  trace  of  its  plane. 

47.  Find  the  traces  of  the  plane  of  the  three  following  points: 

48.  The  H  projection  of  a  line  AB  has  a  =  0,  0,  —  |;  h  =  2i,  0, 
U.  It  lies  in  a  plane  T,  with  TO  =  2f ,  0,  0.  TV  has  1  =  4f ,  1|,  0. 
TH  has  2  =  0,  0,  |.    Find  the  other  projection  of  the  line  AB. 

49.  A  =  0,  —  1,  —  1t¥.  a  plane  T  containing  the  point  has  TO 
=  If,  0,  0.     TH  has  2  at  3f ,  0,  U.    Find  TV. 

Analysis:— Since  the  point  lies  in  the  plane,  a  line  of  the  plane 
may  be  drawn  to  pass  through  it,  and  by  means  of  its  traces,  the 
required  trace  of  the  plane  can  be  found. 

Construction:— Draw  any  line  through  the  point  A,  and  lying  in 
the  plane  T,  preferably  a  vertical  of  the  plane.  Its  H  projection 
will  be  parallel  to  the  G.L.  Where  it  cuts  TH  prolonged  is  the  H 
projection  of  its  H  trace;  the  V  projection  of  its  H  trace  is  on  the 
G.L.  at  the  foot  of  a  perpendicular  through  the  H  projection.  The 
vertical  projection,  of  the  vertical  of  the  plane,  goes  through  the 
vertical  projection  of  the  H  trace  and  the  point  a' .  The  V  trace 
of  T  is  parallel  to  this  line  and  goes  through  TO. 

50.  AB  and  CD  are  two  intersecting  lines.    A  =  0,  —  |,  |  and  B 


EXERCISES 

TV     ?« 


205 


TH 


'U- 


51. 


52. 


ia' 


r 


53. 


54. 


=  2, 2f ,  U;  C  =  f ,  2f ,  If,  and  D*  =  3f ,  -,  0.  A  point  O  lying  in  the 
plane  of  AB  and  CD,  has  o'  =  If,  |,  0.  Find  o  without  using  the 
traces  of  the  plane  of  the  lines. 

To  Follow  Section  51,  Page  43. 

51.  Draw  a  line  through  the  point  A  and  parallel  to  the  plane  T. 

52.  Draw  a  line  through  the  point  A  and  parallel  to  the  plane  T. 

53.  Draw  a  line  through  the  point  A  and  parallel  to  the  plane  T. 

54.  Draw  a  line  through  the  point  A  and  parallel  to  the  plane  T. 

a'  c/ 


55. 


55.  A  is  a  point  in  the  plane  T.    Draw  a  line  through  the  point 
B  and  parallel  to  the  plane  T,  which  is  not  a  horizontal  line. 

To  Follow  Section  52,  Page  44. 

56.  Pass  a  plane  through  the  line  AB  and  parallel  to  the  line  CD. 

57.  Pass  a  plane  through  the  line  AB  and  parallel  to  the  line  CD. 

*  When  a  coordinate  is  left  with  a  — ,  It  means  that  It  Is  to  be  found.  It  Is 
impracticable  to  give  by  coordinates,  four  points  accurately  on  two  Inter- 
secting lines. 


206 


EXERCISES. 


oT 


f 


69.  60. 

58.  Pass  a  plane  through  the  line  AB  and  parallel  to  the  line  CD. 

59.  Pass  a  plane  through  the  line  CD  and  parallel  to  the  line  AB. ' 

60.  Pass  a  plane  through  the  line  AB  and  parallel  to  the  line  CD. 

b 

^pf     a 


a' 

\ 

*' 

P 

d    , 

\ 

V 

^6 

a 

df 

61. 


62. 


63. 


64. 


To  Follow  Section  54,   Page  46. 


61.  Pass  a  plane  through  the  point  O  and  parallel  to  the  lines 
AB  and  CD. 

62.  Pass  a  plane  through  the  point  P  and  parallel  to  the  lines 
AB  and  CD. 

63.  Pass  a  plane  through  the  point  P  and  parallel  to  the  lines 
AB  and  CD. 

64.  Pass  a  plane  through  the  point  P  and  parallel  to  the  two 
lines  AB  and  CD. 

To  Follow    Section  56,  Page  48. 

65.  T  and  S  are  two  planes  with  TV  and  TS  at  45°  to  the  G.L. 


EXERCISES  207 

towards  the  right;  TV  intersects  the  G.L.  at  0,  0,  0;  SV  intersects 
the  G.L.  at  1,  0,  0.  The  H  traces  of  both  intersect  at  3i,  0,  f .  Find 
their  line  of  intersection. 

66.  T  and  S  are  two  planes.  T  is  perpendicular  to  V;  TV 
making  45°  with  the  G.L.  toward  the  left,  cutting  same  at  3\,  0,  0. 
S  cuts  the  G.L.  at  2,  0,  0;  SV  makes  an  angle  of  60°  with  the  G.L. 
towards  the  left,  and  SH,  30°  with  the  G.L.  in  front  of,  and  to  the 
right.    Find  the  line  of  intersection  of  the  two  planes. 

67.  T  and  S  are  two  planes.  TV  is  perpendicular  to  the  G.L., 
cutting  it  at  0,  0,0;  TH  makes  an  angle  of  30°  with  the  G.L.  in 
front  of  and  toward  the  right.  SH  is  perpendicular  to  the  G.L., 
cutting  it  at  3,  0,  0;  SV  makes  an  angle  of  45°  with  the  G.L.,  above 
and  toward  the  left.  Find  the  line  of  intersection  of  the  two 
planes. 

68.  T  and  S  are  two  planes.  TH  and  SH  are  perpendicular 
to  the  G.L.  and  cut  the  latter,  respectively,  at  f ,  0,  0  and  2\,  0,  0. 
Their  V  traces  intersect  each  other  at  a  point  0,  2i,  0.  Find  the 
line  of  intersection  of  the  planes. 

To  Follow  Section  57,  Page  51. 

69.  T  is  a  plane  intersecting  the  G.L.  at  0,  0,  0.  TV  makes  an 
angle  at  30°  with  the  G.L.  and  TH,  45°  with  the  G.L.,  both  toward 
the  right.  A  line  has  both  projections  parallel  to  the  G.L.  and 
goes  through  the  point  A  =  0,  1,  li.  Find  where  the  line  pierces 
the  plane. 

70.  AB  is  a  line,  with  A  =  0,  —  li,  -  f ,  and  B  =  li,  U,  |.  Find 
where  it  pierces  a  plane  T  with  TV  perpendicular  to  the  G.L.  and 
cutting  the  latter  at  2|,  0,  0,  and  TH  making  an  angle  of  30°  with 
the  G.L.  toward  the  left. 

71.  A  line  AB  has  A  =  0,  If,  i,  and  B  =  0,  |,  1.  A  plane  T  has 
TV  and  TH  coinciding  and  at  30°  to  the  G.L.  above  and  toward  the 
left.    Find  where  the  line  AB  pierces  the  plane  T. 

72.  A  plane  T  has  both  traces  parallel  to  the  G  L.    TV  U  inch 


208 


EXERCISES 


6' 

b i 


73. 


74. 


75. 


76. 


above  it,  and  TH,  li  inch  in  front  of  it.    A  line  has  A  =  0,  i,  —  i, 
and  B  =  li,  H,  —  1^.    Find  where  the  line  AB  pierces  the  plane  T. 

To  Follow  Section  £9,  Page  53. 

73.  Through   the  point    O,  pass  a  plane  perpendicular  to  the 
line  AB. 

74.  Through   the   point  O,  pass   a  plane  perpendicular  to  the 
line  AB. 

75.  Through  the   point  O,  pass   a  plane  perpendicular  to   the 
lineAB. 


76.  Through  the  point  O,  pass  a  plane  perpendicular  to  the 
line  AB. 

To  Follow  Section  60,  Page  55. 

77.  Plane  T  is  perpendicular  to  H  and  V  at  0,  0,  0.  A  point  O 
is  at  f ,  f ,  \.    Find  the  distance  from  the  point  O  to  the  plane  T. 

78.  Plane  T  is  perpendicular  to  H,  makes  an  angle  of  20°  with 
V,  cuts  G.L.  at  0,  0,  0.  A  point  O  is  at  |,  f ,  f .  Find  its  distance 
from  the  plane  T. 

79.  T  has  its  V  trace  parallel  to  the  G.L.,  |  inch  above  it,  its  H 
trace  is  parallel  to  the  G.L.,  \  inch  in  front  of  it.  O  is  a  point  at 
0,  —  I,  —  f .     Find  its  distance  from  the  plane  T. 

80.  A  plane  T  is  perpendicular  to  V  and  makes  20°  with  H., 
cutting  the  G.L.  at  0,  0,  0.  A  point  O  is  at  f,  0,  f.  Find  its 
distance  from  the  plane  T. 


EXERCISES  209 

81.  A  plane  T  has  its  V  trace  parallel  to  the  G.L.,  f  inch  below 
H,  its  H  trace  parallel  to  the  G.L.,  f  inch  back  of  V.  O  is  any 
point  on  the  G.L.     Find  its  distance  from  the  plane  T. 

82.  A  plane  T  has  both  traces  parallel  to  the  G.L.  and  coincid- 
ing; and  I  inch  above  it.  A  point  O  is  at  0,  —  |,  f .  Find  its 
distance  from  the  plane  T. 

To  Follow  Section  63,  Page  58. 

83.  A  =  0,  0,  f ;  B  =  U,  U,  U;  C  -  1,  i^,  |;  D  =  A,  I,  -.  Find 
the  angle  between  the  lines  AB  and  CD. 

84.  A  plane  T  has  V  trace  at  45°  with  the  G.L.  above  and 
towards  the  right,  and  H  trace  at  30°  with  the  G.L.  below  and 
towards  the  left.    Find  the  angle  between  the  two  traces. 

85.  A,  B  and  C  are  the  vertices  of  a  triangle.  A  =  0,  |,  li; 
B  =  4,  H,  f ;  C  =  If,  i,  1.  Find  its  true  shape  and  the  angles 
between  the  sides. 

86.  A  =  0,  -  t,  -  i;  B  =  1,  -  f ,  -  i;  C  =  i,  -  1,  -  f.  Find 
the  angle  between  the  lines  AB  and  BC. 

87.  A  =0,-1,-1;  B  =  0,  -  t,  0;  C  =  i,  -  1,  —  |.  Find  the 
angle  between  the  lines  AB  and  BC. 

88.  A  =  0,  If,  —  1;  B  =  0,  f,  0;  C  =  0,  -  i,  -  1.  Find  the 
angle  between  the  lines  A  B  and  B  C. 

89.  Given  a  plane  T  whose  traces  coincide  and  make  an  angle 
of  45°  with  the  G.L.  Choose  any  point  on  the  plane,  not  on  either 
trace,  and  draw  two  lines  lying  in  the  plane  and  making  an  angle 
of  60°  with  each  other. 

90.  Given  a  plane  T  with  TO  =  0,  0,  0.  A  point  1  on  the  V 
trace  =  2i,  —  f ,  0,  and  a  point  2  on  the  H  trace  =  2i,  0,  If.  A  line 
AB  in  it  has  A  =  1,  li,  0,  and  B  =  2,  0,  — .  Draw  a  line  in  the 
plane  making  an  angle  of  90°  with  AB. 

91.  A  line  AB  has  A   =  0,  —  f ,  f ;  B  =  2f ,  |,  —  i.    Pass  a  plane 


210 


EXERCISES 


through  the  line  and  parallel  to  the  G.L.  At  a  point  O,  which 
divides  the  line  AB  in  the  ratio  of  5  to  4,  draw  a  line  making  90° 
with  it  and  lying  in  the  plane  passed  through  it. 

92.  O  =  5,  —  I,  —  i.  It  is  a  point  in  a  plane  T,  which  is  per- 
pendicular to  H  and  makes  an  angle  of  30°  with  V.  Draw  two 
lines  through  O  making  an  angle  of  60°  with  each  other. 

93.  0  =  0,  —  1,-2.  It  is  a  point  in  a  plane  containing  the 
G.L.  draw  two  lines  through  O  and  in  the  plane  making  an  angle 
of  120°  with  each  other. 


y 


94. 


^/ 


/ 


"<^ 


76 


95. 


\^y 


T 


/ 

96. 


aa" 


97. 


To  Follow  Section  66,  Page  59. 

94.  Project  the  line  AB  on  the  plane  T. 

95.  Project  the  line  AB  on  the  plan^  T. 

96.  Project  the  line  AB  on  the  plane  T. 

97.  Project  the  line  AB  on  the  plane  shown. 


of 


rrv 


TH 


h' 


98. 


c/ 


99. 


TH 


TV 


h 
lOO. 


98.    Project  the  line  AB  on  the  plane   T  determined  by   the 
point  C. 


EXERCISES 
99.    Project  the  line  AB  on  the  plane  T. 
lOO.    Project  the  line  AB  on  the  plane  T. 


lOl. 


f^/ 


/ 


g 


t\ 


>^ 


\ 


N 


102. 


103. 


To  Follow  Section  68,  Page  62. 

101.  Find  the  angle  between  the  planes  T  and  S. 

102.  Find  the  angle  between  the  planes  T  and  S. 

103.  Find  the  angle  between  the  planes  T  and  S. 

104.  Find  the  angle  between  the  planes  T  and  S. 


./f     \ 


r 


r 


/ 


t 


v< 


\ 


105. 


106. 


\     / 

<^  \iS> 


107. 


/ 


211 


j^ 


P 


sV^ 


P 


104. 


^/^ 


y 


y 


108. 


105.  Find  the  angle  between  the  planes  T  and  S,  the  latter  be- 
ing determined  by  the  point  O. 

106.  Find  the  angle  between  the  planes  T  and  S. 

107.  Find  the  angle  between  the  planes  T  and  S. 

108.  Find  the  angle  between  the  planes  T  and  S. 

To  Follow  Section  77,  Page  75. 

109.  A  plane  T  has  both  traces  coinciding  and  making  an  angle 


212  EXERCISES 

of  45°  with  the  G.L.  toward  the  right.  Draw  the  projections  of  a 
triangular  pyramid  of  21  inches  altitude,  and  equilateral  triangular 
base  one  inch  on  a  side,  an  edge  of  the  base  in  H  and  plane  of  base 
in  T. 

no.  A  plane  T  has  TV  at  60°  to  the  G.L.  above  and  to  the  right. 
TH  at  45°  to  the  G.L.,  below  and  toward  the  right.  A  base  of  a 
cube  lies  in  it  with  one  corner  in  V  and  a  diagonal  of  the  base  par- 
allel to  V.    The  cube  is  H  inches  on  edge.    Draw  its  projections. 

HI.  Find  the  locus  of  points  equi-distant  from  the  two  points: 
A  =  0,  1,  f ;  and  B  =  If,  H,  H. 

112.  Find  the  locus  of  points  in  space  equi-distant  from  t>he  two 
following  planes,  T  and  S.  Point  1  of  T*=  0,  i,  0,  and  3  =  2i,  i,  0. 
Point  2  =  0,  0,  1,  and  4  =  2i,  0,  1.  Point  1  of  S  =  0,  If,  0,  and  3  = 
2i,  If,  0.     Point  2  =  0,  0,  1,  and  4  =  2^,  0,  1. 

113.  Find  the  locus  of  points  in  space  equi-distant  from  the 
points:     A  =  0,  1,  |,  and  B  =  |,  f ,  i^,  and  C  =  li^.  If,  2. 

To   Follow   Section  83,   Page  82. 

114.  A  =  0,  1^,  i;  B  =  1,  0,  0,;  C  =  If,  If,  i.  Find  the  locus  of 
points  in  space  equi-distant  from  the  lines  AB  and  BC. 

115.  N  =  0,  -  i,  -  If;  P  =  f,  -  U,  -  1;  O  =  If,  -  i,  _  ^. 
Draw  the  locus  of  points  in  space  equi-distant  from  the  lines  NO 
and  OP. 

To  Follow  Section  102,  Page  101. 

116.  MN  is  the  axis  of  an  oblique  cylinder,  with  circular  base 
H  inches  in  diameter,  in  H.  M  =  0,  0,  |;  N  =  2,  If,  li.  The  bases 
are  parallel;  O,  P  and  Q  are  points  on  the  surface;  o'  =  i,  f ,  0;  p' 
=  1,  If,  0;  g  =  2i,  0,  I;  find  their  corresponding  projections;  draw  a 
tangent  plane  to  the  cylinder  at  Q. 

117.  MN  is  the  axis  of  an  oblique  cylinder,  having  a  circular 
base  in  H,  li  inches  in  diameter.     The  bases  are  parallel;  M  =  0, 

*The  odd  numbers  are  points  on  the  V  trace,  the  even  those  on  the  H  trace. 


EXERCISES  213 

0, 1;  N  =  H,  H,  H.  P  is  a  point  on  the  cylinder  at  i,  li,  —  ;  draw 
a  plane  tangent  to  the  cylinder  at  P;  draw  another  plane  normal 
to  the  cylinder  at  the  point  P. 

To  Follow  Section  114,  Page  112. 

118.  Develop  the  cylinder  of  problem  117. 

To  Follow  Section  117,  Page  116. 

119.  A  righ  circular  cone  of  f  inch  diameter  of  base,  If  inches 
altitude,  has  its  axis  parallel  to,  and  equi-distant  from,  V  and  H  in 
the  3rd  angle;  center  of  base  is  at  0,  —  |,  — f.  A  point  P  on  its 
surface  has  its  H  projection  at  i^,  0,  —  1.  Draw  a  plane  tangent 
to  the  cone  at  P. 

120.  An  oblique  cone,  with  circular  base  in  H,  H  inches  in 
diameter,  and  center  on  the  G.L,  at  0,  0,  0,  has  apex  at  li^,  li,  li^. 
A  point  P  on  its  surface  is  at  i^,  — ,  f ;  draw  a  tangent  plane  to  the 
cone  at  P. 

To  Follow  Section  119,  Page  119. 
112.    An   oblique  cone,  with  circular  base   in  H,   H  inches  in 
diameter,  and  center  on  the  G.L.  at  0,  0,  0,  has  apex  at  li\,  H,  1^. 
A  line  MN  has  M  =  1^,  r«,  —  ts;  N  =  2^,  1,-1;  draw  a  tangent 
plane  to  the  cone  and  parallel  to  the  line  MN. 

122.  A  right  circular  cone,  of  f  inch  diameter  of  base  If  inches 
altitude,  has  its  axis  parallel  to,  and  equi-distant  from,  V  and  H, 
in  the  3rd  angle;  center  of  base  is  at  0,  —  f ,  —  |.  A  line  BC  has  B 
=  2,  —  f ,  —  1^;  C  =  21,  —  i,  —  1.  Draw  a  plane  tangent  to  the 
cone  and  parallel  to  the  line  BC. 

123.  A  right  circular  cone  has  base  in  H,  2  inches  in  diameter, 
2i  inches  altitude,  and  center  at  2i,  0,  Ij.  MN  is  a  line  with  M  = 
0,  i,  It,  and  N  =  li,  li,  i.  Pass  a  plane  tangent  to  the  cone 
and  parallel  to  the  line  MN. 

124.  Discuss  the  conditions  of  problem  123  when  (1),  the  given 
line  has  the  same  inclination  to  the  plane  of  the  base  of  the  cone 
that  the  elements  of  the  cone  do;  (2),  when  the  given  line  makes  a 


214  EXERCISES. 

less  angle,  with  the  plane  of  the  base  of  the  cone,  than  the  elements 
do. 

To  Follow    Section  123,  Page  122. 

125.  An  oblique  cone  has  apex  at  0,  If,  2.  The  base  touches  H 
at  the  point  If,  0,  If,  is  a  circle  in  H  projection,  and  its  plane  is 
perpendicular  to  V  and  makes  an  angle  of  30°  with  H.  A  plane  T, 
perpendicular  to  the  axis  of  the  cone,  cuts  the  G.L.  at  f ,  0,  0. 
Find  the  curve  of  intersection  of  the  cone  and  the  plane. 

To  Follow  Section  126,  Page  125. 

126.  Develop  the  cone  of  problem  125. 

To  Follow  Section  150,  Page  161. 

127.  Draw  the  contour  of  an  oblique  helicoid  of  uniform  pitch, 
with  the  following  conditions:  One  helical  directrix,  H  inches  in 
diameter,  second  helical  directrix,  4  inches  in  diameter;  pitch  of 
helices  4  inches;  angle  of  elements  with  the  axis  15°.  Take  the  H 
plane  perpendicular,  and  the  V  plane  parallel  to  the  axis  of  the 
helical  directrices.  Plot  the  curves  of  at  least  three  points  of  the 
generatrix,  in  addition  to  those  in  which  it  touches  the  two  given 
vdirectrices. 

128.  Draw  the  contour  of  a  right  helicoid  of  uniform  pitch,  with 
the  following  conditions:  One  helical  directrix  1^  inches  in 
diameter,  second  helical  directrix  4  inches  in  diameter,  pitch  of 
helices  4  inches.  Take  the  H  plane  parallel  to  the  plane  director, 
and  the  V  plane  parallel  to  the  axis  of  the  helical  directrices. 
Plot  the  curves  of  at  least  three  points  of  the  generatrix,  in 
addition  to  those  in  which  it  touches  the  two  given  directrices. 

To  Follow   Section  159,   Page  168. 

129.  An  oblate  spheroid  rests  on  H  and  touches  V.  The 
minor  axis  is  vertical  and  is  2  inches  long,  touching  H  at  3,  0,  H; 
the  major  axis  is  2i  inches  long.  P  is  a  point  on  the  upper 
surface,  and  its  V  projection  is  at  3f ,  li,  0.  Pass  a  plane  through 
P  and  the  minor  axis.  Show  the  curve  of  intersection  of  the  plane 
with  the  spheroid;  draw  a  plane  tangent  to  the  surface  at  P. 


EXERCISES  .215 

To  Follow  Section  161,  Page  170. 

130.  A  sphere,  2^  inches  in  diameter,  rests  on  H,  with  center 
li  inches  from  V  in  the  1st  angle.  A  point  P,  on  the  front 
surface  of  the  sphere,  has  its  V  projection  |  inch  above  the 
center  and  |  inch  to  the  right.  Draw  a  plane  tangent  to  the 
sphere  at  the  point  P. 

131.  An  ellipsoid  of  revolution  has  a  major  axis  If  inches  long, 
minor,  li  inches.  It  lies  in  the  3rd  angle,  touching  the  H  plane, 
the  major  axis  is  parallel  to  V,  1  inch  from  it,  and  is  perpendicular 
to  H.  A  point  P,  on  the  surface,  has  its  V  projection  f  inch  to 
the  left  of  the  axis,  and  If  inch  below  H.  Draw  a  plane  tangent  to 
the  surface  at  the  point  P. 

132.  A  circle,  1  inch  in  diameter,  revolves  about  a  vertical  axis 
which  touches  H  at  a  point  li,  0,  1|,  thus  generating  a  torus. 
The  center  of  the  circle  is  at  a  distance  of  1  inch  from  the  axis, 
and  the  torus  rests  on  H.  A  point  upon  the  surface  is  at  a  distance 
of  f  inch  from  the  axis  and  li  inches  from  V;  it  may  have  several 
positions,  show  them  and  pass  planes  tangent  to  the  surface  at 
each. 

To  Follow  Section  167,  Page  180. 

133.  A  line  AB  has  A  =  0,  0,  3i;  B  =  2f,  3,  li;  it  revolves  about 
an  axis  perpendicular  to  H,  whose  H  projection  is  the  point  li,  0, 
1|,  thus  generating  a  hyperboloid  of  revolution  of  one  nappe. 
Draw  the  upper  base  generated  by  the  point  B,  and  the  lower, 
generated  by  the  point  A;  draw  the  circle  of  the  gorge,  and  the 
principal  meridian  plane;  draw  through  B  an  element  of  the 
second  generation. 

Locate  a  point  P,  on  the  surface,  at  If,  — ,  If ;  a  point  R  at  I, 
I,  — .     Draw  a  plane  tangent  to  the  surface  at  P. 

Find  the  curve  of  intersection,  with  a  plane  S,  cutting  the  G.L. 
at  31,  0,  0,  and  whose  V  and  H  traces  make  angles  of  30°  with  the 
G.L.  toward  the  right,  above  and  below  the  G.L.,  respectively. 

To  Follow  Section  174,  Page  187. 

134.  Two  oblique  cones,  with  circular  bases  in  H,  intersect  each 


216 


EXERCISES 


other.  One  cone  has  base  2i  inches  in  diameter,  its  center  being 
at  0,  0,  2,  and  apex  at  2i,  31,  f ;  the  other  cone  has  base  3  inches  in 
diameter,  its  center  being  at  2|,  0,  2|,  and  apex  at  |,  2i,  i.  Draw 
the  curve  or  curves  of  intersection  of  the  two  cones,  using  a 
sufficient  number  of  points  to  obtain  a  good  curve. 


135. 


136. 


136.    Prove  that  the  point  C  does  not  lie  on  the  line  AB. 

136.  Prove  that  the  two  lines  AB  and  CD  do  not  lie  in  the 
plane  T. 

137.  A  =  0, 1,  f ;  B  -  I,  U,  I;  P  =  If,  |,  |.  A  plane  T  has  TO 
=  If,  0.  0;  TV  makes  an  angle  of  45"  with  the  G.L.  above  and  to 
the  right;  TH  makes  an  angle  of  45°  with  the  G.L.  below  and  to 
the  right.  Draw,  through  P,  a  plan©  parallel  to  the  line  AB,  and 
perpendicular  to  the  plane  T. 


138. 

the  line  AB  and  distant  1  inch  from  P. 


Draw  a  plane  through 


139.  A  plane  T  has  TO  =  0,  0,  0;  TV  makes  an  angle  of  30°  with 
the  G.L.  above  and  to  the  right;  TH  makes  an  angle  of  45°  with 
with  the  G.L.  below  and  toward  the  right.  A  point  A,  in  the 
plane,  is  at  1,  |,  — .  Through  A  draw  a  line  making  an  angle  of 
30°  with  H. 


140.  A  line  AB  makes  an  angle  of  30°  with  H;  its  V  trace,  A, 
is  at  0,  I,  0;  the  end  B  has  its  H  projection  at  If,  0,  1.  Draw  a 
line  through  the  H  trace  of  AB,  lying  in  H,  and  making  an  angle 
of  45°  with  AB. 


EXERCISES 


217 


141. 

141.  Given  the  two  pulleys  as  shown,  and  in  3rd  angle  projec- 
ion,  and,  assigning  dimensional  values  to  them,  etc.,  find  the 
position  of  the  center,  and  the  direction  of  the  axis  of  an  idler,  of 
any  given  diameter,  which  will  take  the  belt  from  the  left  hand 
pulley  and  deliver  it  to  the  right  hand  one  without  its  slipping  off. 


i 


\ 


€) 


142. 


143. 


142.  Assigning   numerical    values    to   the   edges    of  the  form 
shown,  find: 

(1).    The  angles  between  the  sides  of  the  solid. 
(2) .    The  angles  of  the  bevel  of  the  edges  between  the  different 
sides. 

143.  Find  the  curve  of  intersection  between  the  stem  and  end 
of  the  connecting  rod  as  shown. 


218 


EXERCISES 


144. 

144.  Given  the  *two  way'  flue,  as  shown,  and  assigning  dimen- 
sional values  to  the  form,  develop  each  piece  as  for  sheet  metal 
work. 


145. 


145.  Given  the  form  shown,  which  is  a  *cylindrical  to  square' 
offset,  develop  it  by  triangulation,  as  discussed  in  the  appendix, 
namely,  take  four  triangles,  whose  bases  are  the  edges,  respec- 
tively, of  the  square  opening,  and  whose  apexes  are  in  the 
cylindrical  portion  below.    Between  these,  divide  the  cylindrical 


EXERCISES 


219 


opening  into  a  number  of  parts,  the  bases  of  another  set  of 
triangles  whose  apexes  are  in  the  corners  of  the  square  opening 
above. 


146. 


147. 


146.  Given  the  form  shown,  which  is  a  cylindrical  chimney- 
above  a  square  opening,  and,  assigning  to  it  dimensional  values, 
develop  the  surface,  as  for  sheet  metal  work,  by  the  method  of 
triangulation  described  in  the  appendix,  namely,  take  first,  four 
triangles,  whose  bases  are,  respectively,  in  contact  with  the  four 
lower  edges,  and  apexes  in  the  neck  above;  then  a  series  of 
triangles  whose  bases  are  contiguous  and  form  part  of  the  neck, 
with  sides  touching  and  apexes  at  the  corners  where  the  four 
lower  edges  meet. 


147.  Develop  the  frustum  of  the  conical  figure  shown,  by  the 
method  of  triangulation  discussed  in  the  appendix,  namely,  divide 
the  surface  into  a  number  of  triangles,  whose  bases  are  oppositely 
directed,  and  lie  in  the  upper  and  lower  bases  of  the  form,  respec- 
tively, and  whose  sides  are  touching  one  another. 


220 


EXERCISES 


148. 


148.  Given  the  form  shown,  which  is  analogous  to  the  slope 
sheet  of  a  locomotive  boiler,  and  assigning  dimensional  values  to 
it,  develop  it,  as  for  sheet  metal  work,  by  the  method  of  triangula- 
tion,  discussed  in  the  appendix,  namely,  divide  the  upper  and 
lower  bases  into  a  large  number  of  equal  parts;  let  the  spaces 
between  the  divisions  be  the  bases  of  a  set  of  triangles,  one  set  in 
the  upper  curve,  the  other  set  in  the  lower  curve,  and  the  apexes 
of  each  set  in  the  opposite  curve,  the  edges  of  the  triangles 
coincident. 


149.  Given  the  form  shown,  which  is  a  'round  to  flat'  flue,  and 
assigning  dimensional  values  to  it,  develop  it  as  for  sheet  metal 
work,  by  the  method  of  triangulation  as  described  in  the  appendix 
namely,  take  two  triangles,  with  apexes  at  a  and  d,  respectively 


EXERCISES 


221 


149. 

and  with  bases  6c  and  e/  respectively;  then,  divide  up  the  remain- 
der of  each  base  into  a  large  number  of  parts,  each  one  into  the 
same  number.  Regard  the  divisions  in  each  base  as  the  bases  of 
a  set  of  triangles  with  apexes  in  the  opposite  base,  at  the  points 
of  division  on  that  base. 


INDEX 


PAGE 

Alphabet  of  the  line 26 

Alphabet  of  the  plane 12 

Alphabet  of  the  point 6 

Analysis   defined    24 

Angles,  to  draw  a  line  through 
a  given  point  at  a  given 
angle  to  an  oblique  plane  79 
Angle  of  a  plane  with  a  co- 
ordinate plane,  being 
given,  and  the  trace  with 
that    plane,    to    find    the 

other  trace   68 

Angle  of  a  plane  with  a  co- 
ordinate plane  being 
given,  and  the  trace  with 
the  corresponding  coordi- 
nate plane,  to  find  the  re- 
maining trace 70 

Angle,  to  draw  a  plane  at  a 
given  angle  to  another 
and     through     a     giveij 

line  65 

Angles,   dihedral    3 

Angles  between  lines 58,    59 

Angles,  a  line  making  given 
angles  with  the  coordi- 
nate planes   45 

Angles  a  line  makes  with  both 
coordinate  planes,  to  find 

its  projections 73 

Angles,  lines  lying  in  four  dihe- 
dral angles   25 

Angles    a    line    makes    with    a 

plane   61 

Angles  between  two  planes 62 

Angles  between  two  planes,  by 
change      of      coordinate 

planes    66 

Angles,  given  the  angles  a 
plane  makes  with  both 
coordinate  planes  to  find 

both  traces 72 

Appendix    197 


PAGE 

Approximate  development    198 

Archimedian  spiral  in  classifi- 
cation,        86 

Archimedian  spiral  defined....     88 
Archimedian  spiral  as  the  trace 
of    an     oblique     helicoid 

and  a  plane 163 

Asymptote   defined    93 

Auxiliary  sphere,  used  to  ob- 
tain traces  of  a  plane 
with   coordinate  planes..     72 

Axes  of  reference   8 

Axis  of  a  cone  defined 115 

Axis  of  a  hyperbolic  paraboloid  154 

Base  of  a  cone  defined 115 

Base  of  a  cylinder  defined....     99 

Center  of  projection  3 

Change  of  coordinate  planes 
for    angle    between    two 

planes   66 

Change    of    coordinate    planes 

for  a  line   31,     39 

Change   of   a   coordinate  plane 

with  respect  to  a  plane. 40,  41 
Change    of     coordinate     planes 

with  respect  to  a  point.  17,  18 
Characteristic  name  for  a  cone.  115 
Circle     through     three     given 

points    77 

Circle  as  a  locus 80 

Circle,  in  classification   86 

Circle,  involute  of,   classified..     86 

Circle  defined   87 

Circular,  right  circular  cone  of 
revolution;  cylinder  of 
revolution  91 

Circular,    to    develop    a    right 

circular  cylinder  110 

Circle  as  intersection  of  sur- 
faces of  revolution  hav- 
ing a  common  axis 171 


224 


INDEX 


PAGE 

Circle  of  the  gorge  of  a  surface 

of  revolution  167 

Circle  of  the  gorge  of  a  hyper- 

boloid  of  revolution   ....  175 

Classification  of  lines   86 

Classification  of  surfaces   91 

Cone,  axis  of,  defined 115 

Cone,  base  of,  defined 115 

Cone,  in  classification  91 

Cone,  defined 88,  155,  166 

Cones,  intersection  of  two   187 

Cone,    intersection    of,    and    a 

convolute    191 

Cone,  intersection  of,  and  a  cyl- 
inder    189 

Cone,  intersection  of  any,  by  a 

plane    120 

Cone,  nappe  of,  defined 116 

Cone,  oblique,  defined   115 

Cone,  to  develop   122,  124,  125 

Cone,  to  draw  a  plane  normal 
to,  at  a  point  on  the  sur- 
face      119 

Cone,  to  assume  a  point  on 116 

Cone,  plane  tangent  to  and  par- 
allel to  a  line 119 

Cone,    plane    tangent   to,    at   a 

point  on  surface 116 

Cone,  plane  tangent  to,  through 

a  point  outside   117 

Cone  of  revolution   defined 115 

Cone,  of  revolution,  right  cir- 
cular         91 

Conical  helix  classified   86 

Conical  helix  defined   88 

Conies  defined  86,     87 

Conoid,  in  classification  ....91,  140 
Conoid  defined   155 

Conoid,     intersection     of,     and 

plane   161 

Conoid,  to  assume  a  point  on, 
and  to  draw  a  plane  tan- 
gent at  the  point 156 

Conoid,  right  elliptical,  defined  158 

Conoid,  right  helical  defined...  155 


PAGE 

Consecutive  elements  of  a  sin- 
gle curved  surface  lying 
in  a  tangent  plane  to  the 

surface    98 

Consecutive  positions  of  a  mov- 
ing point  defined  83 

Contour  elements  of  a  solid...  107 

Conventions  13 

Convolute,  in  classification 91 

Convolute  defined 89,  126 

Convolute,  helical,  to  represent  129 

Convolute,  helical  defined 126 

Convolute,  helical,  to  develop . .  137 
Convolute,    helical,    to    draw   a 
plane  tangent  to  and  par- 
allel to  a  line 135 

Convolute,    helical,   intersection 

of,  with  a  plane 132 

Convolute,  intersection  of,  with 

a  cone 191 

Convolute,  intersection  of,  with 

a  cylinder   191 

Convolute,  to  assume  a  point 
on,  and  to  draw  a  plane 

tangent  to   131 

Coordinate  axes  8 

Coordinate  planes,  line  parallel 

to  both   19 

Coordinate  plane,  line  perpen- 
dicular  to    20 

Coordinate  planes  of  projection 

defined    1 

Coordinate  planes  of  projection, 
angles  a  line  makes  with 
both,  to  find  its  projec- 
tions       73 

Coordinate  planes  of  projection, 
angles  a  plane  makes 
with  both,  to  find  its  pro- 
jections         72 

Coordinate  planes  of  projection, 
change  of,  for  angle  be- 
tween two  planes    66 

Coordinate  planes  of  projection, 

change  of,  for  a  line.  .31,     39 
Coordinate  planes  of  projection, 
change   of,    with    respect 
to  a  plane  40,     41 


INDEX 


225 


PAGE 

Coordinate  planes  of  projection, 
cliange  of,  with  respect 
to  a  point 17.    18 

Coordinate  planes  of  projection, 

distance  of  a  point  from.6,  IS 

Coordinate  planes  of  projection, 
line  making  given  angles 
with    45 

Coordinate  planes  of  projection, 

revolution  of   4 

Coordinate  planes  of  projection, 

trace  of,  with  a  line.  .21,     22 

Curvature,  curve  of  double, 
formed  by  the  motion  of 
a  plane  intersecting  in 
successive  elements  of...     84 

Curvature,  development  of  sur- 
face of  single   108 

Curvature  line  moving  tangent 

to  a  curve  of  double 126 

Curvature,  a  right  line  tan- 
gent to  a  curve  of  single, 
defined  94 

Concourse,  point  of,  defined 143 

Corresponding,  use  of  word. ...     14 

Corresponding   projection   of   a 

point  on  a  line 15 

Corresponding  trace  of  project- 
ing plane  of  a  line 15 

Curve  defined  as  envelope  of  a 

moving  line 84 

Curve  of  double  curvature, 
formed  by  the  motion  of 
a  plane  intersecting  in 
successive  elements  of 
the  curve  84 

Curve  of  intersection  of  a  plane 

and  a  surface  107 

Curve,  the  meridian  curve  of  a 
hyperboloid  of  revolution 
is  a  hyperbola  178 

Curve,  meridian  curve  of  a  sur- 
face of  revolution 167 

Curve,  projection  of,  defined...     92 

Curve,  a  right  line  tangent  to  a 
curve  of  single  curvature 
defined  94 

Curve,  tangency  of  two,  defined     94 


PAGE 

Curve,  to  draw  a  tangent  to  an 

irregular  plane,    95 

Curved  lines  classified  86 

Curved,  double  curved  lines  de- 
fined, also  single   84,     88 

Curved,  double,  surfaces  class- 
ified,       91 

Curved,  double,  surfaces  defined    89 

Curved,  double,  surfaces  of  rev- 
olution       91 

Curved,  single,  surfaces,  two 
consecutive  elements  ly- 
ing in  a  tangent  plane  to     98 

Curved,  single  surface,  the  cyl- 
inder defined  98 

Curved,  single,  surface,  develop- 
ment      108 

Curved,  single,  surface,  inter- 
section with  a  surface  of 
revolution   194 

Curved,  single,  surface,  to  as- 
sume point  on 99 

Curved,  single,  surface,  to  as- 
sume an  element  on....  100 

Cycloid,   in  classification 86 

Cylinder  defined    88,  98,  166 

Cylinder,  to  develop  a 112 

Cylinder,  to  develop  a  right  cir- 
cular,      110 

Cylinder,  in  classification 91 

Cylinder,    intersection    of    cone 

and   189 

Cylinder,  intersection  of.  and  a 

convolute    191 

Cylinder,  intersection  of  two..  186 

Cylinder,  intersection  of  a  right 

circular,  and  a  plane 108 

Cylinder,    normal    plane  to,    at 

a  point,    105 

Cylinder,  plane  normal  to,  also 

parallel  to  a  line 106 

Cylinder,  projecting,  of  a  solid.  107 

Cylinder,     right     circular,     in 

classification    91 

Cylinder,  to  draw  a  plane  tan- 
gent to,  and  parallel  to  a 
line  103 


226 


INDEX 


PAGE 

Cylinder,  to  draw  a  plane  tan- 
gent to,  at  a  point  on 101 

Cylinder,    to    draw    a    tangent 
plane    through     a    point 

outside 102 

Declivity,  line  of  greatest 57 

Definition    86,     87 

Degree  of  an  equation,  the  maxi- 
mlum    number    of    times 
a   line   may   cut  a  curve    93 
Descriptive  geometry,  defined  . .       1 
Descriptive     geometry,     proper 

study  of   24 

Develop,   to,  any   cone  in   gen- 
eral    124 

Develop,  to,  an  oblique  cone  . .  125 
Develop,  to,  an  oblique  cylinder  112 
Develop,  to,  a  helical  convolute  137 
Develop,     to,     a  right  circular 

cone  122 

Develop,  to,  a  right  circular  cyl- 
inder       110 

Develop,  to,  a  surface  of  single 

curvature    lOS 

Development,  approximate   198 

Diametral    planes    of    a   hyper- 
bolic   paraboloid    150 

Dihedral  angles   3 

Dihedral  angles,  lines   connect- 
ing points  in  different  . .     25 

Directrix,  defined 87 

Directrix,    helical    123 

Distance  of  point  from  coordi- 
nate planes  6,    18 

Distance  from  a  point  to  a  line    70 
Division,  the  projection  of  any 
division  of  an  angle  be- 
tween lines 59 

Double  curvature,  line  moving 

tangent  to  a  curve  of  . . .  126 
Double     curvature,     curve     of, 
formed  by  the  motion  of 
a   plane  intersecting     in 
successive     elements     of 

curve   84 

Double  curved  lines  classified  86 
Double  curved  lines  defined  . .     84 


PAGE 

Double   curved   surfaces   classi- 
fied        91 

Double      curved      surface      de- 
fined     89,  165 

Double    curved    surface,    plane 

tangent  to   165 

Double  curved  surfaces  of  revo- 
lution in  classification...     91 

Eccentricity,    defined    80 

Element,     to     assumie,     on     a 

cone 116 

Element,  to   assume,  on  single 

curved  surface   100 

Element   of   path   traced   by   a 

moving  point  83 

Element  of  tangency  defined...     98 
Elements  consecutive,  lying  in 
a     plane     tangent     to     a 
single   curved   surface. . .     98 
Elements  of  contour  defined...  107 

Ellipse,  defined  87 

Ellipse  in  classification 86 

Ellipse,  as  a  locus 80 

Ellipsoid  of  revolution  in  classi- 
fication      91 

Ellipsoid  of  unequaled  axes  de- 
fined       92 

Elliptical  cylinder  defined 99 

Elliptical  hyperboloid  in  classi- 
fication  140 

Elliptical  hyperboloid  defined..  140 

Elliptical  cone,  oblique 117 

Elliptical     conoid,     right,     de- 
fined    158 

End  plane,  line  parallel  to   . . .     20 
End  plane,  trace  of  a  line  with 

23,     24 

End  vertical  plane 1,      3 

End     vertical     plane,     ground 

line  13 

Envelope,  curve  defined  as  the, 

of  a  moving  line 84 

Epicycloid  in  classification 86 

Epicycloid  defined   87 

Equiangular  spiral  in  classifica- 
tion         86 


INDEX 


227 


PAGE 

First  and  third  angle  projection 

discussed    197 

Focus  defined   80 

Front  vertical  plane   1,      3 

Geometry,  descriptive  defined..  1 
Geometry,  projective,  defined..  2 
Generation    of     double    curved 

surfaces    166 

Generation,  method  of,  of  sur- 
faces         91 

Generation  of  a  cone  defined  . .  115 

Generatrix  of  a  cylinder.  .\ 99 

Generatrix  defined    87 

Generic  name  for  cone 115 

Gorge,  circle  of  the,  of  a  hyper- 

boloid  of  revolution  ....  175 
Gorge,  circle  of  the,  of  a  sur- 
face of  revolution  167 

Gorge  plane,  of  hyperboloid  of 

revolution  175 

Gorge    plane    of    a    surface    of 

revolution 167 

Graphical  locii   81 

Ground  line  3 

Ground   line  with  end  vertical 

plane    13 

Ground  line  as  projection  of  V 

and  H  planes  20 

Gussett,  method  of  develop- 
ment  198 

Hi  plane,  supplementary IS 

Helical      conoid,      intersection 

with,  and  a  plane 161 

Helical,  conoid,  oblique 156 

Helical,  conoid,  right,  defined..  155 
Helical    convolute,    intersection 

with  a  plane 132 

Helical  convolute  defined 126 

Helical  convolute,  to  develop. .  137 
Helical  convolute,  plane  tangent 

to  and  parallel  to  a  line  135 
Helical  convolute,  plane  tangent 
to,  through  a  point  out- 
side    133 

Helical  convolute,  to  assume 
point  on  and  draw  a  tan- 
gent plane  to 131 


PAGE 

Helical  convolute,  to  represent  129 

Helicoid,  in  classification   91 

Helicoid,  right  and  oblique,  etc., 

defined 158 

Helicoid,  oblique,  classified 140 

Helicoid,     oblique,     to     assume 

point  on   161 

Helicoid,  oblique,  the  point  of 
tangency    of      a      plane 

with  161 

Helicoid,  oblique,  trace  of  with 
a  plane,   an   archimedian 

spiral    163 

Helicoid,  right,  classified 140 

Helicoid,  right,  illustrated 162 

Helix,  angular  pitch 128 

Helix,  classified 86 

Helix,  conical,  defined  88 

Helix,   defined    87,126 

Helix,  linear  pitch   128 

Horizontal  plane 1,      3 

Horizontal  of  a  plane    37 

Horizontal  projecting  line 5 

Hyperbola  in   classification   ...     86 

Hyperbola,   defined    87 

Hyperbola,  as  a  locus 80 

Hyperboloid,  elliptical  classified  140 

Hyperboloid  of  revolution,   the 

mieridian  curve  of 178 

Hyperbolic  paraboloid,  its  axis.  154 
Hyperbolic  paraboloid,  in  class- 
ification    91,  140 

Hyperbolic    paraboloid     defined 

141,  146 

Hyperbolic  paraboloid,  diame- 
tral planes  of, 150 

Hyperbolic  paraboloid,  a  plane 
tangent  to  and  through 
a  line 155 

Hyperbolic  paraboloid,  plane 
tangent  to,  at  a  point  on 
the  surface   149 

Hyperbolic       paraboloid,         to 

assume  a  point  on,   .147,  154 


228 


INDEX 


PAGB 

Hyperbolic  paraboloid,  its  pro- 
jections and  sections  by- 
coordinate  planes,  and 
tangent  plane  at  any 
point    151 

Hyperbolic  paraboloid,  theorem 

12    143 

Hyperbolic  paraboloid,  theorem 

13    144 

Hyperbolic  paraboloid,  the  ver- 
tex of 154 

Hyperbolic  spiral,  in  classifica- 
tion         86 

Hyperboloid   of  revolution,     in 

classification 91,  140 

Hyperboloid  of  revolution,  de- 
fined     90,  173 

Hyperboloid  of  revolution,  in- 
tersection by  a  plane. . . .  178 

Hyperboloid  of  revolution,  me- 
ridian curve  is  a  hyper- 
bola       178 

Hyperboloid     of    revolution,  to 
assume      point     on     and 
draw  tangent  plane  to . . .  180 
Hypocycloid,   in  classification..     86 

Hypocycloid  defined   87 

Indeterminate   projections  of  a 

line 15 

Indeterminate     position     of     a 

plane   11 

Intersecting,  angle  between  two, 

lines 58,     59 

Intersection  of  two  cones   ....  187 
Intersection  of  a  cone  and  con- 
volute      191 

Intersection   of   a   cone   and   a 

cylinder 189 

Intersection   of  a  cylinder  and 

convolute    191 

Intersection  of  two  cylinders..  186 
Intersection  of  a  cone  and  plane  120 
Intersection  of  an  oblique  heli- 
cal conoid  with  a  plane. .  161 
Intersection  of  a  helical  convo- 
lute and  a  plane 132 


PAGE 

Intersection  of  a  plane  and  hy- 
perboloid of  revolution..   182 

Intersection,    line    as,    of    two 

planes    38 

Intersection,    line    of,    of    two 

planes  47,     48 

Intersection     of     two      bodies 

bounded  by  plane  faces. .   184 

Intersection  of  two  pyramids..  185 

Intersection  of  surfaces   183 

Intersection  of  a  surface  and  a 

plane    107 

Intersection  of  a  single  curved 
surface  and  surface  of 
revolution 194 

Intersection,    point    of,    of    two 

lines  16 

Intersection  of  a  right  circular 

cylinder  and  a  plane   . . .   108 

Intersection  of  surfaces  of  revo- 
lution, general  cases    . . .  192 

Intersection  of  surface  of  revo- 
lution by  a  plane   172 

Intersection  of  two  surfaces  of 

revolution  171 

Intersection  of  surfaces  of  revo- 
lution having  a  common 
axis    192 

Intersection  of  surfaces  of  revo- 
lution, axes  in  the  same 
plane    192 

Intersection  of  surfaces  of  revo- 
lution, axes  not  in  the 
same   plane    194 

Intersection,  as  a  trace 11 

Involute  of  a  circle,  in  classifi- 
cation       86 

Line,  alphabet  of 26 

Line,  making  given  angles  with 

a  coordinate  plane   45 

Line,  angles  made  with  both  co- 
ordinate planes   73 

Lines,  angle  between 58,    59 

Line,    angle   it   makes    with    a 

plane   61 

Lines   classified    86 

Lines,  double  curved,  defined. .     84 


INDEX 


229 


PAGE 

Line,      change     of     coordinate 

planes   for   31 

Lines  as  determining  a  plane. .     11 
Line,  distance  from  a  point  to. .     70 

Line  of  greatest  declivity 57 

Lines,  notation  for 13 

Lines,  projecting  2 

Lines,  projecting  planes  of   . . .     15 
Lines  generating  a  surface  by- 
motion  88 

Lines,  horizontal  projecting.  .4,      5 
Line,  indeterminate  projections 

of    15 

Line,    the   intersection    of   two 

planes  47,     48 

Lines,  point  of  intersection  of 

two   IG 

Lines,  length  of,  by  change  of 

coordinate  planes    32 

Line,  length  of  by  revolution. .     32 
Line,  path  of  moving  point  ...     83 
Line,   normal  to  a  surface   de- 
fined       97 

Line,   to   draw,     parallel    to    a 

plane  through  a  point. . .     43 
Line,    parallel    to    both    coordi- 
nate planes   19 

Line  parallel  to  a  plane 20 

Line  perpendicular  to  a  coordi- 
nate plane  20 

Line  perpendicular  to  a  plane. 52,  53 
Line,     plane     through,     and    a 

point    37 

Line,    plane   through   one,   and 

parallel  to  another 44 

Lines,  the  common  perpendicu- 
lar of  two  non-intersect- 
ing lines,  67,     68 

Line,  to  project  on  any  oblique 

plane    59 

Line  connecting  points  in  dif- 
ferent angles    25 

Line,  to  designate  a  point  on..     15 
Lines,    perpendicular    and    one 
parallel    to    a   coordinate 

plane    57 

Line,  projecting 5 


PAGE 

Lines,  projections  of,  fully  de- 
termined        14 

Lines,    corresponding    trace    of 

the  projecting  plane  of. .     15 
Line,     revolution     of    a    point 

about 27,     31 

Lines,  right,  defined 84 

Lines,  single  curved,  defined...     84 
Line,   tangent   to   a   curve,    de- 
fined     93,     94 

Line,  traces  of 20,    21,     22 

Line,  trace  of  with  any  plane.     51 
Line,    trace    of    with    an    end 

plane  23,     24 

Line,  traces  of,  to  contain  a...     37 
Line,  the  traces  of  its  project- 
ing plane  14 

Line,  rule  to  find  the  traces  of. 

with  a  surface 108 

Lines,  vertical  projecting   4,  5 

Linear  pitch  of  a  helix 128 

Locii,  defined 80 

Locus,  the  circle,  ellipse,  hyper- 
bola, parabola  as  a 80 

Locus    of     points     equidistant 

from  two  planes 81 

Locus     of     points     equidistant 

from  three  points   81 

Locus  of  projecting  perpendic- 
ulars         14 

Logarithmic  spiral  in  classifi- 
cation       86 

Meridian  curve  of  a  surface  of 

revolution  167 

Meridian  curve  of  a  hyperbo- 
loid  of  revolution  is  a  hy- 
perbola      178 

Meridian  plane  of  a  surface  of 
revolution  at  a  point  is 
perpendicular  to  the  tan- 
gent   plane   through    the 

point 170 

Nappe,  of  a  cone  defined 116 

Non-intersecting  lines,  the  com- 
mon perpendicular  to  . . .     68 
Non-intersecting    lines    as    ele- 
ments of  a   warped   sur- 
face      141 


230 


INDEX 


PAGE 

Normal  to  a  curve,  defined 94 

Normal  to  a  surface,  defined. ..     97 
Normal  plane  to  a  cone  through 

a  point  on, 119 

Normal    plane    to    a    cylinder, 

through  a  point  on..  105,  106 
Normal  plane  to  a  surface  de- 
fined       97 

Notation   13 

Oblique  cone  defined    115 

Oblique  cone,  to  develop 125 

Oblique  cone,  elliptical   117 

Oblique  conoid,  in  classifica- 
tion         91 

Oblique  cylinder,  defined   99 

Oblique  cylinder,  to  develop...   112 
Oblique,  conoid,  helical,  defined  156 

Oblique  helicoid  classified 140 

Oblique  conoid,  helical,  inter- 
section with  a  plane....   161 

Oblique  helicoid,  defined 15S 

Oblique  helicoid,    illustrated...   162 
Oblique     helicoid     of     varying 

pitch,  classified 140 

Oblique   helicoid,   to   assume   a 

point  on, 161 

Oblique  helicoid,  point  of  tan- 

gency  of  any  plane  with.   164 
Oblique  helicoid.   trace  with   a 
plane,     an     archimedian 

spiral   163 

Oblique  hyperbolic  paraboloid..  154 
Oblique  plane,  change  of  coor- 
dinate planes  for 40 

Oblique  plane,  to  draw  a  line 
through  a  point  at  a 
given  angle  to  79 

Oblique  plane,  to  draw  a  pyra- 
mid with  base  in 75 

Parabola,   in   classification 86 

Parabola,  defined  87 

Parabola,  as  a  locus, 80 

Paraboloid,  hyperbolic,  its  axis  154 
Paraboloid,  hyperbolic,  in  class- 
ification    91,  140 


PAGB 

Paraboloid,  hyperbolic,  de- 
fined    141,  14o 

Paraboloid,  hyperbolic,  diame- 
tral planes  of   150 

Paraboloid,  hyperbolic,  tangent 

plane  to,  through  a  line.   155 

Paraboloid,  hyperbolic,  tangent 

plane  to,  at  a  point 149 

Paraboloid,  hyperbolic,  to  as- 
sume a  point  on 147,  154 

Paraboloid,  hyperbolic,  projec- 
tions of,  sections  by  coor- 
dinate planes  and  tan- 
gent plane  to   151 

Paraboloid,  hyperbolic,  the  ver- 
tex      154 

Paraboloid,  hyperbolic,  theoremi 

12    143 

Paraboloid,  hyperbolic,  theorem 

13    144 

Paraboloid     of     revolution,     in 

classification   91 

Paraboloid,  variable,  defined...     92 
Paraboloid    of    revolution,    de- 
fined     167 

Parallel,  line,  to  a  plane 20,     43 

Parallel,  line,  to  both  coordi- 
nate planes    19 

Parallel,  line,  to  an  end  plane,.     20 
Parallel,    plane    tangent    to    a 
cone    and    parallel    to    a 

line  119 

Parallel  planes,  traces  of  with 

a  third  plane   39 

Parallel,  plane  normal  to  a  cyl- 
inder   and    parallel   to   a 

line  106 

Partial  penetration  of  one  form 

with  another 185 

Path,   element   of,  traced   by  a 

moving  point   S3 

Path  as  a  locus 80 

Penetration,  complete  or  partial 

of  one  form  with  another  185 
Perpendicular,       line,       to       a 

plane   52,     56 

Perpendicular,  line,  to  coordi- 
nate plane   20 


INDEX 


231 


PAGE 

Perpendicular,  a  plane  through 
a  line,  perpendicular  to  a 

given  plane 79 

Perpendicular,  the  common, 
to     two     non-intersecting 

lines    67,     68 

Perpendicular,  a  plane,  through 

a  point,  to  a  line 53 

Perpendicular,  a  plane  tangent 
to  surface  of  revolution 
is  perpendicular  to  the 
meridian    curve    through 

the  point   170 

Perpendicular,  projecting,  as  a 

locus    14 

Perspective,  defined   2 

Pitch,  angular,  of  a  helix 128 

Pitch,  helicoid  of  uniform,  de- 
fined     158 

Plane  the  alphabet  of, 12 

Plane,   angle   between   two . .  62,     66 
Plane  at  given  angle  with  an- 
other  through   a   line   in 

the  latter   65 

Plane,  in  classification   91 

Plane,  defined  88 

Plane,  the  angles  with  both  co- 
ordinate planes  being 
given,  to  find  its  traces.     72 

Plane  of  coordinate  axes 8 

Plane,  end  vertical,  front  verti- 
cal, horizontal   1^      3 

Plane,   indeterminate  position..     11 

Plane,  notation  for   13 

Plane,  given  to  draw  a  pyra- 
mid with  base  in   75 

Planes,    change    of    coordinate, 

with  respect  to  a  line. 31,     39 
Planes,  change  of  both  coordi- 
nate planes  with  respect 

to  a  plane 40,    41 

Planes,  revolution  of  coordinate      4 
Planes  of  projection,  defined ...       1 
Planes    of   projection    for    per- 
spective           2 

Planes,  diametral,  of  a  hyperbo- 

loid  of  revolution   150 


PAGE 

Plane,  gorge  of,  hyperboloid  of 

revolution  175 

Plane,  gorge,  of  surface  of  rev- 
olution   167 

Plane,  intersection  of  any,  cone 

by  a  120 

Plane,    intersection    of,    and    a 

conoid,  helical    161 

Plane,    intersection    of,    and    a 

right  circular   cylinder. .  108 

Plane,  intersection  of  and  a  hy- 
perboloid of  revolution..  168 

Plane,  intersecting  itself  in  suc- 
cessive elements  of  a 
curve  of  double  curva- 
ture         84 

Plane,  intersecting  any  surface  107 

Plane,  intersecting  any  surface 

of   revolution    172 

Plane,    line    as    intersection    of 

two  38 

Plane,    line   of    intersection    of 

two    47,     48 

Plane,  angle  a  line  makes  with,     61 

Planes,  angle  a  line  makes  with 
both  coordinate,  to  find 
its  projections 73 

Planes,  line  making  giving  an- 
gles  with   coordinate....     45 

Planes,  coordinate,  line  perpen- 
dicular to  20 

Plane,  the  horizontal  of 37 

Plane,  line  through  a  point  at  a 

given  angle  to 79 

Plane,  through  one  line  parallel 

to  another  line 44 

Plane,  line  parallel  to  20 

Plane,  to  draw  a  line  parallel 

to,  through  a  point 43 

Plane,     through     a    line     and 

point    37 

Plane,     line     perpendicular    to 

any  52 

Plane,  line  perpendicular  to,  is 
.perpendicular  to  any  line 
in  it 56 

Plane,  through  a  line  perpen- 
dicular to  a  plane 79 


232 


INDEX 


PAGE 

Plane,  to  project  a  line  on  any,     59 

Plane,  the  vertical  of 37 

Plane,   normal  to   a  cone  at  a 

point    119 

Plane,  normal  to  a  cylinder  at 

a   point    105,  106 

Plane,    normal    to    a    cylinder 

through  a  point  outside.  106 
Plane,  normal  to  a  surface,  de- 
fined       97 

Plane,  through  three  points ...     35 
Plane,  to  assume  a  point  in  a . .     38 
Planes,    locus    of    points    equi- 
distant  from   two 81 

Plane  through  a  point  parallel 

to  two  lines 48 

Plane  through  a  point  perpen- 
dicular to  a  line 53 

Plane,  as  projected  on  coordi- 
nate planes   11 

Planes,  projecting,  of  a  line...     15 
Plane,  revolution  of  supplemen- 
tary vertical   17 

Plane,  tangent   to  a  cone 116 

Plane,    tangent    to    a    cylinder 

and  parallel  to  a  line 103 

Plane,  tangent  to  a  cylinder  at 
a  point  on,  point  out- 
side     101,  102 

Plane,  tangent,  defined  97 

Plane,   tangent  to   a   cone   and 

parallel  to  a  line 119 

Plane,     tangent     to     a     cone, 

through  a  point  outside.  117 

Plane,  tangent  to  a   conoid  at 

a  point  on  the  surface..  156 

Plane,     tangent    to     a     double 

curved  surface   165 

Plane,  tangent  to  a  helical  con- 
volute     131 

Plane,  tangent  to  a  helical  con- 
volute and  parallel  to  a 
line  135 

Plane,  tangent  to  a  helical  con- 
volute through  a  point 
outside    133 


PAGE 

Plane,  tangent  to  a  hyperbolic 
paraboloid  and  through  a 
line  155 

Plane,   point  of  tangency   with 

oblique  helicoid    164 

Plane,  tangent  to  a  hyperbolic 
paraboloid  at  a  point  on 
surface    149 

Plane,  tangent  to  hyperboloid 
of  revolution  at  a  point 
on  surface   180 

Plane,     tangent     to     a     sphere 

through  a  line 170 

Plane,  tangent  to  surface  of 
revolution  at  a  point  on 
surface 167 

Planes,  tangent,  to  warped  sur- 
faces     149 

Plane,  trace  of  a  line  with  co- 
ordinate     21,     22 

Plane,  traces  of  11 

Plane,  traces  of  not  limited...     16 

Plane,  one  trace  given  and  an- 
gle with  corresponding 
plane  to  find  remaining 
trace  70 

Plane,  one  trace  given  and  an- 
gle with  plane  of  that 
trace  to  find  the  corre- 
sponding trace   68 

Plane,   traces  of,   to   contain  a 

line  37 

Plane,   corresponding  traces  of 

projecting  plane  of  a  line.  .15 

Plane,  traces  of  parallel  planes 

with  any  other  plane...     89 

Plane,  trace  of  a  line  with  any,     51 

Point,  the  alphabet  of 6 

Point,     change     of     coordinate 

planes  with  respect  to.  17,  ]8 

Point,  circle  through  three....     77 

Point  of  concourse,  defined....   143 

Point,  to  assume,  on  a  cone. . .  116 

Point,  plane  tangent  to,  through 

a  point  outside 117 

Point,  on  a  conoid,  to  draw  a 

tangent  plane   156 

Points,  consecutive 83 


INDEX 


233 


PAGE 

Point,  to  designate  on  a  line. . .  15 
Point,  distance  from,  to  a  line.  70 
Point,     distance     from     coordi- 
nate planes   6,  18 

Point,    element   of   path   traced 

by  a  moving 83 

Point,   to   assume  on   a  helical 

convolute    131 

Point  on  line  in  different  angles  25 

Point,  through  given,  to  draw 
a  line  making  given 
angles  with  any  plane...     79 

Point,  to  draw  a  line  parallel 

to  a  plane  through, 43 

Point,  locus  of,  equally  distant 

from  two  planes   81 

Point,  to  assume,  on  an  oblique 

helicoid   161 

Point,  to  assume,  on  a  hyperbo- 
loid  of  revolution  and 
draw  tangent  plane 180 

Point,  to  assume,  on  hyperbolic 
paraboloid,  and  draw 
plane   tangent   to    149 

Point,  of  intersection     of     two 

lines    16 

Points,    locus     of,     equidistant 

from  three  points   81 

Point,  notation  for   13 

Point,  to  assume  in  a  plane. ...     38 

Points,  plane  through  three. . .     35 

Point,  plane  through  line  and..   37 

Points,  to  pass  a  plane  parallel 

to  two  lines  through, ...     46 

Point,  plane  through,  perpen- 
dicular to  a  line 53 

Point,  the  projections  of 5,      6 

Point,  corresponding  projections 

of,  on  a  line 15 

Point,  the  projections  of,  on  a 
common  perpendicular  to 
the  G.  L 6 

Point,  position  of,  determined 
by  its  projections,  theo- 
rem 1 7 

Point,  revolved  about  a  line  27,    31 


PAGE 

Point  of  sight   2 

Point,     to    assume,     on    single 

curved  surface   99 

Point,  to  assume  on  surface  of 

revolution  167 

Point  of  tangency  defined 94 

Point,  approximate,       of       tan- 
gency of  a  curve  and  its 

tangent   96 

Principal   meridian   curve  of  a 

surface  of  revolution   . .  .   167 

Prisms,  intersection  of  two 184 

Project,  a  line  on  any  plane. ..     59 
Projecting  cylinder  of  any  solid  107 
Projecting  lines,  horizontal,  ver- 
tical, etc 2,  4,      5 

Projective  perpendiculars,  as  lo- 
cus        14 

Projecting  plane  of  a  line 15 

Projection,  center  of 2,      3 

Projection,    corresponding    pro- 
jection  of   a  point  on   a 

line  15 

Projection  of  forms  3,      6 

Projections,    to   find    the,    of   a 
line  when  angles  with  co- 
ordinate planes  are  given     73 
Projections  of  a  line  fully  deter- 
mined         14 

Projections   of   a   line    indeter- 
minate       15 

Projections  of  a  point   5,       6 

Projections   of  a    point     deter- 
mine its  position 7 

Projective  geometry  2 

Proof,    form    of    17 

Pyramids,  intersection  of  two     172 
Pj^ramid,    to    draw,    with    base 

in  given  oblique  plane  . .     75 
Reciprocal  spiral  86 

Rectangular  hyperbolic  parabo- 
loid    154 

Rectified    curve    95 

Right  lines  defined  84 

Right      conoid,     in      classifica- 
tion       91 


234 


INDEX 


PAGE 

Right  circular  cone  of  revolu- 
tion      91 

Right     circular      cylinder,      in 

classification    91 

Right  line  tangent  to  another, 

defined 94 

Right  line  tangent  to  curve  of 

single  curvature 94 

Right  cylinder  defined 99 

Right  circular  cylinder,  to  de- 
velop    110 

Right  circular  cylinder,  inter- 
section by  a  plane 108 

Right  helicoid,  classified   140 

Right   helical   concoid,    defined, 

right  conoid 155 

Right  circular  cone,  to  develop.  122 

Right  helicoid,  defined 159 

Right  conoid,  elliptical,  defined  158 
Revolution,  cone  of,  defined...  115 
Revolution,  hyperboloid  of,  de- 
fined     173 

Resolution,       hyperboloid       of, 

classified  140 

Revolution,  hyperboloid  of,  in- 
tersection by  a  plane 182 

Revolution,  hyperboloid  of,  me- 
ridian    curve    of,     is    a 

hyperbola 178 

Revolution,  hyperboloid  of,  to 
assume  a  point  on  and 
draw  a  tangent  plane  to.  182 

Revolution,  hyperboloid  of,  has 
two  systems  of  genera- 
tion     176 

Revolution,   length    of  line   by,     32 
Revolution,     paraboloid  of,  de- 
fined    167 

Revolution  of  coordinate  planes,      4 
Revolution    of    point    about    a 

line    27,     31 

Revolution,  surface  of 89 

Revolution,    surface    of,    classi- 

fed    91 

Revolution,  surface  of,  defined.  166 
Revolution,  surface  of,  intersec- 
tion of,  general  case 192 


PAGE 

Revolution,  surfaces  of,  inter- 
secting when  having  a 
common  axis 171 

Revolution,  surfaces  of,  inter- 
secting, axes  in  same 
plane    192 

Revolution,  surfaces  of,  inter- 
secting, axes  not  in  same 
plane    194 

Revolution,  surface  of,  intersec- 
tion by  any  plane 172 

Revolution,  surface  of,  intersec- 
tion with  single  curved 
surface   194 

Revolution,  surface  of,  me- 
ridian curve  of,  defined.  167 

Revolution,  surface  of,  plane 
tangent  at  a  point  is  per- 
pendicular to  the  meri- 
dian plane  through  the 
point 170 

Revolution,     surface  of,     plane 

tangent  to.  at  a  point. . . .  167 

Revolution,  surface  of,  to  as- 
sume a  point  on 167 

Roulettes,  classified,  defined  86,     87 

Rule,  for  intersection  of  plane 

and  surface  107 

Rule,   piercing  point  of  a  line 

and  surface  108 

Rule,  tangent  plane  to  a  sur- 
face         97 

Ruled  surfaces,  classified 91 

Ruled  surfaces,  defined  88 

Ruled  surface  of  revolution,  in 

classification   91 

Secants,  curve  of,  in  classifi- 
cation         86 

Secant  line,  approaching  a  tan- 
gent as  a  limit, 93 

Secant  plane  of  a  cylinder,  as  a 

base    99 

Secant  plane,  approaching  a  tan- 
gent plane  as  a  limit 97 

Sight,  point  of 2 

Single  curved  lines,  classified.     86 

Single  curved  lines,   defined ...     84 


INDEX 


235 


PAGE 

Single  curvature,   line   tangent 

to  a  curve  of 94 

Single  curved  surfaces,  classi- 
fied        91 

Single  curved  surfaces,  defined.     88 
Single  curved  surface,  develop- 
ment    108 

Single  curved  surface,  the  cylin- 
der, defined   98 

Single  curved  surface,  two  con- 
secutive elements  of  lying 

in  a  tangent  plane 98 

Single  curved  surface,  intersect- 
ing a  surface  of  revolu- 
tion     194 

Single  curved  surface,  to  as- 
sume a  point  on 99 

Sinusoid  curve  in  classifica- 
tion         8C 

Solution  of  problems  defined..     24 

Sphere,  in  classification,    91 

Sphere,  defined  16G 

Sphere,   auxiliary,   for  locating 

plane,  traces  given 72 

Sphere,      plane      tangent      to, 

through  a  line 170 

Spirals,   defined    86 

Spiral,  archimedian,  defined...     88 
Spiral,   archimedian,   the  trace 
of  an  oblique  helicoid  and 

a  plane  163 

Supplementary  Hi  plane 1 

Supplementary  vertical  plane. .     17 

Surfaces   classified    91 

Surfaces,  double  curved,  ,  de- 
fined     165 

Surfaces,  intersection  of 183 

Surfaces,  method  of  genera- 
tion         91 

Surfaces,   generated   by   motion 

of  a  line 88 

Surfaces,  curve  of  intersection 

of  and  a  plane   107 

Surface,  normal  plane  to,  de- 
fined         98 

Surface,  tangent  plane  to,  de- 
fined         97 


FAQE 

Surfaces  of  revolution. 89 

Surfaces  of  revolution,  classi- 
fied       91 

Surfaces  of  revolution,  with 
common  axis,  tangent  or 
intersect  in  a  circle  ....  171 

Surfaces  of  revolution  defined,  166 

Surfaces  of  revolution,  intersec- 
tion of,  general  case 192 

Surfaces  of  revolution,  inter- 
section of,  with  common 
axis    192 

Surfaces  of  revolution,  inter- 
section of,  with  axes  in 
same  plane  192 

Surfaces  of  revolution,  inter- 
section by  a  plane 172 

Surfaces      of      revolution,     to 

assume  a  point  on 167 

Surfaces  of  revolution,  tangent 

plane  to,  at  a  point 167 

Surfaces  of  revolution,  plane 
tangent  to  is  perpendicu- 
lar to  the  meridian  plane 
through  point  of  contact  170 

Surface,  rule  to  find  trace  of  a 

line  with    108 

Surfaces,  ruled,   defined    88 

Surfaces,  single  curved  classi- 
fied       91 

Surfaces,  single  curved,  the  cyl- 
inder defined 88 

Surface,  single  curved,  devel- 
opment of   108 

Surface,  single  curved,  two  con- 
secutive elements  lying 
in  tangent  plane  to   ... .     98 

Surfaces,  single  curved,  inter- 
secting a  surface  of  revo- 
lution     194 

Surface,  single  curved,  to  as- 
sume point  on 100 

Surface,   warded,   defined 89 

System  of  generation  of  a  hy- 
perbolic  paraboloid,    . .  .     145 

Tangent,  in  classification  86 

Tangent  line  to   a   curve,   two 

curves  tangent    94,     95 


236 


INDEX 


PAGE 

Tangent  line,  point  of 96 

Tangent  line,  to  line,  defined..     93 

Tangent  line,  the  limit  of  a  se- 
cant         93 

Tangent  lines,  projected  as  tan- 
gents         94 

Tangent    line    of    surfaces    of 

revolution  171 

Tangent    plane,    to    cone    and 

parallel  to  a  line 119 

Tangent  plane  to  cone  at  point 

on   116 

Tangent  plane  to  cone  through 

point  outside 117 

Tangent  plane  to  conoid 156 

Tangent  plane  to  helical  con- 
volute      131 

Tangent  plane  to  helical  convo- 
lute parallel  to  a  line...  135 

Tangent  plane  to  helical  convo- 
lute, through  a  point  out- 
side      133 

Tangent  plane,  to  cylinder,  par- 
allel to  a  line 103 

Tangent  plane  to  a  cylinder  at 

a  point  on  101 

Tangent    plane    to    a    cylinder 

through  a  point  outside. .  102 

Tangent  plane  to  double  curved 

surface    165 

Tangent  plane,  contains  con- 
secutive elements  of 
single  curved  surface  ...     98 

Tangent  plane  to  hyperboloid  of 

revolution    180 

Tangent  plane  to  hyperbolic 
paraboloid  through  a 
line 155 

Tangent  plane  to  hyperbolic 
paraboloid  at  a  point  on 
surface  149,  151 

Tangent    plane,    to    a    sphere, 

through  a  line 170 

Tangent  plane  to  surface  of 
revolution  at  a  point  on 
surface    167 

Tangent  plane,  to  locate  traces 

of    105 


PAGE 

Tangent  planes  to  warped  sur- 
faces    149 

Tangency,  point  of,  defined  ...     94 

Tangency,  approximate  point  of, 

a  curve  and   its   tangent    96 

Theorem    1.      Projection    of    a 

point 7 

Theorem  2.     Line,  as  the  trace 

of  its  projecting  plane. . .     14 

Theorem  3.  Point  of  intersec- 
tion of  two  lines   16 

Theorem   4,   5.     Line  in   plane 

and  line  parallel  to  plane    34 

Theorem  6.     Traces  of  parallel 

planes    39 

Theorem  7.  Line  perpendicu- 
lar to  a  plane 52 

Theorem  8.  Lines  perpendicu- 
lar and  one  parallel  to  co- 
ordinate plane   57 

Theorem    9.      Projections   of   a 

curve    92 

Theorem  10.     Lines  tangent  in 

projection   94 

Theorem  11.     Plane  tangent  to 

single  curved  surface. ...     98 

Theorem  12.  Hyperbolic  para- 
boloid projected  on  plane 
directer   143 

Theorem-  13.  Section  of  hyper- 
bolic paraboloid  144 

Theorem  14.  Trace  of  oblique 
helicoid  an  archimedian 
spiral   163 

Theorem  15.  Plane  tangent  to 
surface  of  revolution  per- 
pendicular to  meridian 
plane   170 

Theorem  16.  Surface  of  rev- 
olution tangent  when 
having  common  axis 171 

Theorem  17.  Hyperboloid  of 
revolution,  two  systems 
of  generation 176 

Theorem  18.  Meridian  curve 
of  hyperboloid  of  revolu- 
tion is  a  hyperbola 178 

Theorem,  form  of  proof  to 17 


INDEX 


237 


PAGE 

Third  and  first  angle  discussed  197 
Three  points,  plane  through...     35 

Torus,  in  classification   91 

Traces  of  a  line 20,21,     22 

Traces   of   a    line    in    different 

angles    25 

Traces  of  a  line  with  an   end 

plane  23,    24 

Trace,  corresponding,  of  the 
projecting     plane     of     a 

line 15 

Trace     of     a     line    with    any 

plane   51 

Trace,   rule   to   find,   of  a  line 

with  a  plane 108 

Trace  of  a  plane 11 

Trace  of  a  plane  to  contain  a 

line 37 

Trace,  to  locate  the,  of  a  tan- 
gent plane   105 

Trace  of  a  plane  given  and 
angle  with  that  plane  to 
find     its      corresponding 

trace 68 

Trace  of  plane  given  and  angle 
with  corresponding  plane 
to  find  other  trace 70 

Traces  of  a  plane  are  not  lim- 
ited       16 

Traces  of  parallel  planes  with 

third  pliane  39 

Traces   of   plane   perpendicular 

to  a  line 52 


PAGE 

Transcendental        curves,       in 

classification    86 

Triangulation,  a  method  of  de- 
velopment      198 

Trigonometric  curves,  in  classi- 
fication      86 

Trochoids,  classified  86 

Trochoids,  defined 87 

Uniform   pitch  of  helicoid    de- 
fined    158 

Variable  ellipsoid,  in  classifica- 
tion      91 

Variable  paraboloid,  in  classifi- 
cation      91 

Variable  paraboloid,  defined ...  92 
Vertex    of    hyperbolic    parabo- 
loid     154 

Vertical  of  a  plane 37 

Vertical  projecting  lines 4 

Vertical  plane,  front,  end....l,  3 

Vertical  projecting  line 5 

Vertical    revolution    of    supple- 
mentary, plane 17 

Warped  surfaces,  classified 91 

Warped  surfaces,  defined  ...89,  139 

Warped  surfaces,  tangent  planes 

to 149 

Windschief  lines  defined 139 

Windschief  lines  as  directrices 

of  a  surface  of  revolution  173 

Zone  method    of    development.  198 


Short-title  Catalogue 

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15 


$5  00 

1  00 

1  50 

1  50 

6  00 

2  00 

2  50 

4  00 

1  25 

1  00 

1  50 

1  25 

1  00 

1  00 

2  00 

1  00 

2  50 

1  25 

2  50 

3  00 

1  50 

1  00 

MEDICAL. 

*  Abderhalden's  Physiological  Chemistry  in  Thirty  Lectures.     (Hall  and 

Defren.) 8vo, 

von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) 12mo, 

Bolduan's  Immune  Sera 12mo, 

Bordet's  Studies  in  Immunity.     (Gay).     (In  Press.) 8vo, 

Davenport's  Statistical  Methods  with  Special  Reference  to  Biological  Varia- 
tions  16mo,  mor. 

Ehrlich's  Collected  Studies  on  Immunity.     (Bolduan.) 8vo, 

*  Fischer's  Physiology  of  Alimentation Large  12mo, 

de  Fursac's  Manual  of  Psychiatry.      (Rosanoff  and  Collins.).. .  .Large  12mo, 

Hammarsten's  Text-book  on  Physiological  Chemistry.     (Mandel.) Svo, 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry .  .Svo, 

Lassar-Cohn's  Practical  Urinary  Analysis.      (Lorenz.) 12mo, 

Mandel's  Hand-book  for  the  Bio-Chemical  Laboratory 12mo, 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.      (Fischer. )..12mo, 

*  Pozzi-Escot's  Toxins  and  Venoms  and  their  Antibodies.     (Cohn.),  .  12mo, 

Rostoski's  Serum  Diagnosis.      (Bolduan.) 12mo, 

Ruddiman's  IncompatibiHties  in  Prescriptions Svo, 

Whys  in  Pharmacy 12mo, 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Omdorflf.)  Svo, 

*  Batter  lee's  Outlines  of  Human  Embryology 12mo, 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students Svo, 

*  Whipple's  Tyhpoid  Fever Large  12mo, 

WoodhuU's  Notes  on  Military  Hygiene 16mo, 

*  Personal  Hygiene 12mo, 

Worcester  and  Atkinson's  Small  Hospitals  Establishment  and  Maintenance, 
and  Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small 
Hospital 12ino,     1  25 


METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis Svo,     4  00 

BoUand's  Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  used 

in  the  Practice  of  Moulding 12mo, 

Iron  Foimder 12mo, 

"  "  Supplement 12mo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

*  Iles's  Lead-smelting 12mo, 

Johnson's    Rapid    Methods  for   the  Chemical   Analysis  of  Special  Steels, 

Steel-making  Alloys  and  Graphite Large  12mo, 

Keep's  Cast  Iron Svo, 

Le  Chatelier's  High-temperature  Measurements.     (Boudouard — Burgess.) 

12mo, 

Metcalf 's  Steel.     A  Manual  for  Steel-users 12mo. 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.).  .  12mo, 

Ruer's  Elements  of  Metallography.     (Mathewson) Svo. 

Smith's  Materials  of  Machines 12mo, 

Tate  and  Stone's  Foundry  Practice 12mo, 

Thurston's  Materials  of  Engineering.     In  Three  Parts Svo, 

Part  I.      Non-metallic  Materials  of  Engineering,  see  Civil  Engineering, 
page  9. 

Part  II.     Iron  and  Steel Svo,     3  60 

Part  III,  A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents Svo,     2  50 

XJlke's  Modem  Electrolytic  Copper  Refining Svo,     3  00 

West's  American  Foundry  Practice 12mo,     2  60 

Moulders'  Text  Book 12mo,     2  60 

16 


3  00 

2  60 

2 

60 

1 

00 

3 

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2 

60 

3 

00 

2 

60 

3 

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2 

00 

2  50 

1 

00 

2 

00 

8  00 

MINERALOGY. 


Baskerville's  Chemical  Elements.     (In  Preparation.). 

Boyd's  Map  of  Southwest  Virginia Pocket-book  form. 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo, 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo. 

Butler's  Pocket  Hand-book  of  Minerals 16mo,  mor. 

Chester's  Catalogue  of  Minerals 8vo,  paper, 

Cloth. 

*  Crane's  Gold  and  Silver 8vo, 

Dana's  First  Appendix  to  Dana's  New  "System  of  Mineralogy".  .Large  8vo, 
Dana's  Second  Appendix  to  Dana's  New  "  System  of  Mineralogy." 

Large  8vo, 

Manual  of  Mineralogy  and  Petrography 12mo, 

Minerals  and  How  to  Study  Them 12mo, 

System  of  Mineralogy Large  8vo,  half  leather, 

Text-book  of  Mineralogy 8vo, 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo, 

Eakle's  Mineral  Tables 8vo, 

Eckel's  Stone  and  Clay  Products  Used  in  Engineering.      (In  Preparation). 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor. 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) 12mo, 

*  Hayes's  Handbook  for  Field  Geologists 16mo,  mor. 

Iddings's  Igneous  Rocks 8vo, 

Rock  Minerals 8vo, 

Johannsen's  Determination  of  Rock-forming  Minerals  in  Thin  Sections.  8vo, 

With  Thumb  Index 

*  Martin's  Laboratory     Guide    to    Qualitative    Analysis    with    the    Blow- 

pipe  12mo, 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo, 

Stones  for  Building  and  Decoration 8vo, 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper. 

Tables  of   Minerals,    Including  the  Use  of  Minerals  and  Statistics  of 

Domestic  Production 8vo, 

*  Pirsson's  Rocks  and  Rock  Minerals 12mo, 

*  Richards's  Synopsis  of  Mineral  Characters 12mo.  mor. 

*  Ries's  Clays:  Their  Occurrence,  Properties  and  Uses 8vo, 

*  Ries  and  Leighton's  History  of  the  Clay-working  Industry  of  the  United 

States 8vo, 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo, 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks.  .  .,.,,,,,..  .8vo, 


MINING. 

*  Beard's  Mine  Gases  and  Explosions Large  12mo.  3  00 

Boyd's  Map  of  Southwest  Virginia Pocket-<book  form,  2  00 

*  Crane's  Gold  and  Silver 8vo,  5  00 

*  Index  of  Mining  Engineering  Literature 8vo,  4  00 

*  8vo,  mor.  5  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects 12mo,  1  00 

Eissler's  Modem  High  Explosives 8vo,  4  00 

Goesel's  Minerals  and  Metals:  A  Reference  Book 16mo,  mor.  3  00 

Ihlseng's  Manual  of  Mining 8vo,  5  00 

*  Iles's  Lead  Smelting 12mo,  2  50 

Peele's  Compressed  Air  Plant  for  Mines 8vo,  3  00 

Riemer's  Shaft  Sinking  Under  Difficult  Conditions.     (Coming  and  Peele).8vo,  3  00 

*  Weaver's  Military  Explosives 8vo,  3  00 

Wilson's  Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 12mo,  2  50 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation 12mo,  1  25 

17 


$2  00 

1  50 

4  00 

3  00 

1  00 

1  25 

5  00 

1  00 

2  00 

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12  50 

4  00 

1  00 

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3  00 

1  25 

1  50 

5  00 

5  00 

5  00 

60 

4  00 

5  00 

50 

1  00 

2  50 

1  25 

5  00 

2  50 

2  00 

2  00 

SANITARY   SCIENCE. 

Association  of  State  and  National  Food  and  Dairy  Departments,  Hartford 

Meeting,  1906 8vo, 

Jamestown  Meeting,  1907 8vo, 

*  Bashore's  Outlines  of  Practical  Sanitation 12mo, 

Sanitation  of  a  Country  House 12mo, 

Sanitation  of  Recreation  Camps  and  Parks 12mo, 

Folwell's  Sewerage.      (Designing,  Construction,  and  Maintenance.) 8vo, 

Water-supply  Engineering 8vo, 

Fowler's  Sewage  Works  Analyses 12mo, 

Fuertes's  Water-filtration  Works 12mo, 

Water  and  Public  Health 12mo, 

Gerhard's  Guide  to  Sanitary  Inspections 12mo, 

*  Modem  Baths  and  Bath  Houses 8vo, 

Sanitation  of  Public  Buildings 12mo, 

Hazen's  Clean  Water  and  How  to  Get  It Large  12mo, 

Filtration  of  Public  Water-supplies 8vo, 

Kinnicut,  Winslow  and  Pratt's  Purification  of  Sewage.     (In  Preparation.) 
Leach's  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Mason's  Examination  of  Water.     (Chemical  and  Bacteriological) 12mo, 

Water-supply.      (Considered  principally  from  a  Sanitary  Standpoint). 

8vo, 

*  Merriman's  Elements  of  Sanitary  Enigneering 8vo, 

Ogden's  Sewer  Design 12mo, 

Parsons's  Disposal  of  Municipal  Refuse 8vo, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis 12mo, 

*  Price's  Handbook  on  Sanitation 12mo, 

Richards's  Cost  of  Cleanness 12mo, 

Cost  of  Food.     A  Study  in  Dietaries 12mo, 

Cost  of  Living  as  Modified  by  Sanitary  Science 12mo, 

Cost  of  Shelter 12mo, 

*  Richards  and  Williams's  Dietary  Computer 8vo, 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo, 

*  Richey's     Plumbers',     Steam-fitters',    and     Tinners'     Edition     (Building 

Mechanics'  Ready  Reference  Series) 16mo,  mor. 

Rideal's  Disinfection  and  the  Preservation  of  Food 8vo, 

Sewage  and  Bacterial  Purification  of  Sewage 8vo, 

Soper's  Air  and  Ventilation  of  Subways 12mo, 

Turneaure  and  Russell's  Public  Water-supplies 8vo, 

Venable's  Garbage  Crematories  in  America 8vo, 

Method  and  Devices  for  Bacterial  Treatment  of  Sewage 8vo, 

Ward  and  Whipple's  Freshwater  Biology.     (In  Press.) 

Whipple's  Microscopy  of  Drinking-water 8vo, 

*  Typhoid  Fever Large  12mo, 

Value  of  Pure  Water Large  12mo, 

Winslow's  Systematic  Relationship  of  the  Coccaceae Large  12mo, 


MISCELLANEOUS. 

Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo.  1  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,  4  00 

Fitzgerald's  Boston  Machinist 18mo,  1  00 

Gannett's  Statistical  Abstract  of  the  World 24mo,  75 

Haines's  American  Railway  Management 12mo,  2  50 

*  Hanusek's  The  Microscopy  of  Technical  Products.     (Win ton) 8vo,  5  00 

18 


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50 

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25 

4 

00 

2  00 

2 

00 

2 

00 

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50 

1 

50 

1 

00 

1 

00 

1 

00 

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00 

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50 

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00 

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50 

4 

00 

4 

00 

2 

50 

5 

00 

2 

00 

3 

00 

3 

50 

3 

00 

1 

00 

2 

50 

Jacobs's  Betterment    Briefs.     A    Collection    of    Published    Papers    on    Or- 
ganized Industrial  Efficiency 8vo, 

Metcalfe's  Cost  of  Manufactures,  and  the  Administration  of  Workshops.. 8 vo, 

Putnam's  Nautical  Charts gvo, 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute  1824-1894. 

Large  12mo, 

Rotherham's  Emphasised  New  Testament Large  8vo. 

Rust's  Ex-Meridian  Altitude.  Azimuth  and  Star-finding  Tables 8vo. 

Standage's  Decoration  of  Wood.  Glass.  Metal,  etc 12rHO, 

Thome's  Structural  and  Physiological  Botany.     (Bennett) 16mo, 

Westermaier's  Compendium  of  General  Botany.     (Schneider) 8vo, 

Winslow's  Elements  of  Applied  Microscopy 12mo, 


HEBREW   AND   CHALDEE    TEXT-BOOOKS. 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  mor.     5  00 

Green's  Elementary  Hebrew  Grammar 12mo,     1  25 


S3 

50 

5 

00 

2  00 

3 

00 

2 

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19 


<iAA^ 


UNIVEESITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 

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expiration  of  loan  period. 


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